There seems to be a persisitent curiosity among admirers of Celtic art with regard to the meanings behind various knots, symbols, and designs. This curiosity is fueled by a wealth of information on the subject, much of which is either slanted, misinformed, or even downright bogus. Just try "Googling" Celtic Symbols and their meanings and see what you get!! (and of course here I am, adding to it...)  Of course this doesn't mean there are no legitimate sources of information to be found out there - just be aware as you're searching, and always consider the source.

The basic truth is the Celts were not ones for writing things down so any meanings they may have attached to their designs will probably never be known. There are a few symbols or designs which can be interpreted with reasonable accuracy based on what we know about similar ones found in other cultures. We can also venture an educated guess here and there based on what we do know about Celtic culture. The rest is open to pure speculation.
That said, here is a brief compilation of the most common symbols and designs along with their generally accepted interpretations. It starts with general explanations of four basic art forms, then lists more specific designs, including ones used in my Folk Wheels. Feel free to agree or disagree. Schools of thought are as numerous and varied as the stones of Ireland but, again, the simple truth is whatever intent or symbolic meaning the original artists might have attached to their works died with them.
Renown Celtic artist George Bain called the circle "man's first step in art". It seems reasonable to assume the creation of the spiral soon followed. The Celts were not the first or only culture to employ these symbols. They were not even the first to use them in Britain. Spirals carved into a large stone at Newgrange in Ireland date back to 2500 BC which, according to anthropological studies, predates the arrival of Celtic speaking people in Ireland by alomost 2000 years. But the Celts were among the first to elaborate upon these primitive symbols and transform them into art, creating designs that were uniquely theirs. These designs became increasingly complex, culminating around the 7th century AD (a time period sometimes referred to as the "Ultimate La Tene"). As with many cultures, both before and after them, early Celts likely used these symbols to represent the sun and/or their pagan sun god, Taranis. It's possible that such concepts as eternity and continuity may have also been attached. Any further interpretations would be purely speculative. For a little more info, scroll down to Spirals - Folk Wheel image.
Knotwork Interlacings
These are the most popularly recognizable forms of Celtic design, and the most contentious when it comes to interpretation. Truth is, there is no "Dictionary of Celtic Knots", no source based in historical fact that provides a list of knot designs and their respective meanings. Scottish art teacher and author George Bain ("Celtic Art: The Methods of Construction", 1951) suggested that knotwork interlacings could be symbols of continuity, an interpretation which has since become widely accepted among artists and craftspeople. It seems a perfectly reasonable assumption. Is it what the original artists intended? Only they know for sure, and they're not talking.....just smiling secretively in their tombs.

Key and Step Patterns
I've seen these described as "spirals in straight lines", which I suppose is one way of interpreting them. Some art historians propose that the labyrinth and maze patterns found in Greek and Roman art were probable sources of inspiration. Similar patterns and designs exist in a multitude of cultures the world over, from Asia to Africa to the Americas, but again, it was the Celts who raised this art form to another level. Distinctly geometric in form, these simple yet mesmerizing designs reached their apex in the mid-first millennium AD upon the pages of manuscripts like the Book of Kells and the Lindisfarne gospels. What they meant, if they meant anything at all, is lost to history. Perhaps, as some have proposed, they represented some kind of spiritual path.....or maybe they were just an interesting way to fill up a space.

It's a common misconception that early Celts were not skilled or refined enough as artists to illustrate life forms in a realistic manner.
Early pagan laws strictly forbade copying the works of the Creator, which is why no "realistic" versions of plants, animals and humans exist in early Celtic art. Instead, limbs, necks, ears, chins, and so on were either simplified or exaggerated and interlaced with one another to create designs which could represent various life forms without endangering their spirits or insulting the One who created them first. They are also among the most complex and difficult to create of all Celtic designs. Artist Aiden Meehan ("Celtic Borders", 1999, pg.9) states, "animal and plant motifs...offer the most variety and interest. They are also the least often attempted, as being just too difficult, [and]...are all the more intriguing for that." As far as the meaning of any specific zoomorphic design is concerned, one must delve into the murky annuls of Celtic mythology and search out the symbolic meaning of that particular life form. Should you decide to conduct such a search be forewarned: an enormous amount of information exists out there - some reasonable, some outlandish, nearly all of it speculative. Find what seems to be the most generally accepted innterpretation and go with that.
As in many ancient pagan systems of belief the sun was revered by the Celts as a source of life-giving light and warmth. Discs displaying a simple cross within a circle like the one shown above have been found at archeological sites throughout Europe and Britain, some simple, some more elaborately decorated. It is thought that these were sometimes placed at shrines or sacred sites as offerings to Taranis, the Celtic sun god. They were likely worn as talismans as well. The symbol itself is considered to be the prototype for the Christian Celtic High Cross.

Following is a list of designs found in my Folk Wheels* along with a brief paragraph on each summarizing either what is generally known or what I've been able to garner out of the research I've done. None of it is set in stone (no pun intended) or entirely complete. All of it is open to criticism/correction, and most of it is hopefully objective (ie, not a reflection of any specific belief or theology) The designs are all original except for "Inverurie Horses" (created by George Bain) and to some extent "Three Hares", an ancient symbol which I simply elaborated on.

They are listed alphabetically according to design name.

*Not every wheel/design is included here, just the ones I'm most commonly asked about.

There is a school of thought that gives significance to the direction of the spirals: sunwise (or clockwise) circling is thought to signify blessing or good luck. To move in this direction is to be in harmony with the earth. Anti-sunwise has been said to represent anything from negative energy to chaos to spell working. The numbers of whorls or arms are also significant: three for the power of three (see below) or the Holy Trinity; four for the four directions, seasons, or elements (earth, air, fire, water). Spirals both sunwise and anti-sunwise found together in equal numbers are considered to be in balance. 
This symbol is actually Neolithic in origin (perhaps even older), not Celtic, but the Celts clearly adapted it to their own use. It fell in line with their religious precepts - that all important things come in threes: body-mind-spirit, birth-growth-death/rebirth, underworld-earth-otherworld, past-present-future, ancestors-faerie-dieties (aka the Three Kindreds)...just to name a few, and that all life moves in eternal cycles. It has been a staple of Celtic design for three millennia. Image at left is from a stone carving found at Newgrange in Ireland.
TRIQUETRA (aka Trinity Knot)
This simple knot in the shape of a triangle is another representation of the Celtic belief in (and respect for) the power of three. Following the introduction of Christianity it came to also represent the Holy Trinity.
It's not likely this symbol/mythological entity is entirely of Celtic origin, but at some point he was certainly adopted into their culture. It's possible the symbol had its beginnings in ancient Rome; Greco/Roman representations of Bacchus and Dionysis definitely display some similarities. Within Celtic mythology he is a spirit of continuity and reincarnation, always appearing with the first buds of spring, remaining throughout the summer, then disappearing as the last leaves of autumn fall - but always with the promise that he will return again in spring, thus enacting the continuous ebb and flow of life. It falls to reason he may have also symbolized fertility.
There are actually at least two versions of this popular motif. The first, which I call the "classic" version, depicts a pot from which vines emerge and intertwine with one another. Sometimes birds and other creatures can be found among the leaves. This style is found within the pages of manuscripts like the Book of Kells and on Pictish stones in Scotland. The second "neo" version is a stylized tree displaying roots, trunk and branches, sometimes enclosed within a circle, sometimes with the roots or branches (or both) intertwining. Both are viable representations of an ancient concept which is not strictly Celtic in origin but certainly Celtic in principle. The tree has its roots embedded within the earth, reaching down into the underworld. Its base and trunk rest upon the surface of the earth, while its branches reach for the heavens, allowing it to exist on all three planes. In autumn its leaves turn brown and fall, lending it a skeletal, deathly appearance. In spring it comes back to life to thrive an grow through summer, then succumbs to the approach of winter once again, completing the cycle of life. It is a link between worlds; a symbol of strength, longevity, and continuity; a source of wisdom. Trees were revered by the Celts and perceived as living beings imbued with magical powers and godlike characteristics. Whenever land was cleared for a settlement one great tree was always left standing. In Ireland it was called "crann bethadh". Chieftans were inaugurated there, and any number of other ceremonies or important events were likely held beneath its branches. One of the greatest insults an opposing tribe could inflict upon their enemy was to cut down their tree, an outrage punishable by the highest penalties.
One school of thought proposes that the rowan tree is the original or traditional Celtic tree of life. To the best of my knowledge no hard evidence exists to support this. It is true that trees in general were revered by the Celts and, more specifically, their religious leaders, the Druids. According to Druidic lore the rowan symbolizes protection. A sprig of rowan placed over the door of one's cottage will keep out evil; berries strung like pearls were sometimes worn around the neck for the same purpose. It also represents the third month in the Druidic Tree Calendar, corresponding to the period between January 21 and February 17 in the Roman canlendar.
As with the Green Man, this is an ancient symbol and not likely of Celtic origin, but it too falls in line with many Celtic precepts, the power of three being foremost. It's possible they associated the hare with the coming of spring; it certainly could have been a symbol of fertility. At one time it was thought this particular design might have been an alchemic symbol for tin, but that has since been disproved. In some places it's been used as an architectural adornment alongside the Green Man,  hinting that it may have been associated with him somehow. It's interesting to note that although all three hares appear to have two ears each, there is actually only one ear per head.
The stag was a symbol of strength, power, leadership, even divinity. It was also considered to have prophetic qualities. The design used in my Folk Wheel would not be considered a classic zoomorphic design because the forms are not interconnected by knotwork and are a little too lifelike, but it's a pleasing design nevertheless.
Horses were highly valued by the Celts. A man might well have been judged by the quality of the horses he kept and by how well he treated them. Evidence exists that shows a good horse was sometimes treated with the same respect and reverence due a chieftan. They were symbols of virility, strength, and stamina. "Inverurie Horses" is a copy of a design created by George Bain, which was inspired by a Pictish image of a horse he found carved into a stone in Inverurie, Scotland. It is also commonly thought to represent the three "Great Horse" goddesses of Celtic mythology: "Epona" (Gaulish), "Macha" (Irish), and "Rhiannon" (Welsh). The design is widely used by craftspeople and marketers alike, and is found on everything from tee shirts and blankets to glassware and pottery.

Most people are familiar with the Claddagh ring, worn either on the right hand with the heart turned outward to mean the wearer is "fancy free", or on the left hand with the heart turned inward to mean the wearer is "spoken for". Not everyone is familiar with the story behind it...
A young man named Richard Joyce from the village of Claddagh near Galway, Ireland was out fishing one day when a band of foreign pirates happened by. Thinking that he and his crew mates might fetch a good price on the slave market, the pirates captured them all and sailed off for richer waters. Richard was sold to a wealthy Moorish goldsmith who taught him the trade, and as the years went by Richard became quite skilled at the craft - enough to eventually be allowed to create some of his own designs. He fashioned a ring for the girl he had left behind, the one he'd intended to marry, with a heart for love held by two hands for friendship, and a crown on top for loyalty. Eventually, no longer in need of his service, his master released him, whereupon Richard immediately fled back to Galway and the little village he once had called home, carrying the ring he had made in his pocket. To his surprise and great joy, he found his love still waiting. They were promptly wed, and the ring he placed on her hand was the one he had made just for her.
The design has ever since been an Irish symbol of love, enduring friendship, and eternal loyalty.

Like the Claddagh, this too is a symbol of love, but it hails from Scotland and is most commonly worn as a brooch. The linked hearts stand for love and the crown is for loyalty - such as is felt for Mary Queen of Scots. The word "luckenbooth" comes from the "locked booths" from which they (and other more costly items) were sold that once lined the Royal Mile in Edinburgh. It is traditionally exchanged upon a couple's betrothal, then given to their first child at birth.

Also known as Brighid's or Brigit's Cross, it is commonly known as an Irish Christian symbol in honor of Brigid of Kildare, one of the patron saints of Ireland, though like many Celtic-Christian symbols it seems to have parallels with paganism. The cross itself could have derived from the pagan sunwheel. Usually fashioned of rushes or straw, it is traditionally made on the first day of February or La Fheile Bhride (St. Brigid's Feast Day), which happens to coincide with the pagan feast day Imbolc. There was also a pagan Goddess named Brigid associated with healing, fire, and holy wells, and as tradition has it the crosses are usually hung in homes and kitchens as protection against fire and evil.
The story of the Christian St. Brigid goes as follows: an important pagan chieftan was dying. Christians in his household sent for Brigid to tak to him about Christ, and hopefully convince him to convert before breathing his last so that his soul would pass peacefully into Heaven. Delerious with fever he wanted nothing to do with her talk of Christ at first. Quietly, she bent down and picked up a handful of rushes off the floor and began weaving them together into the shape of a cross. He became curious and finally asked what she was doing. She continued to weave, explaining softly while she did about how Christ died on the cross. By the time she had finished her work the chieftan had agreed to convert, and was baptized just in time.
Since then the little cross of rushes has been venerated in Ireland.

To some the thistle might be nothing more than a thorny weed. To the people of Scotland, it is a symbol of national strength, unity and freedom. 
The reason for this is as follows...
In the thirteenth century AD an invading force from the north (possibly Norway) landed upon the shores of Scotland, intent upon victory. Thinking to surprise the clansmen while they slept they chose to wait for nightfall. And to help make their approach even stealthier they decided to remove their footware. As they drew near the sleeping clansfolk they soon discovered they were not the only ones protected by the cover of darkness, for hiding in the grass was a thorny surprise. One Norseman's bare foot stepped upon a thistle and he let out a cry of pain so loud it alerted the Scots of their approach.Clansmen arose and armed themselves, and when the battle was over the Norse were defeated.
The first use of the thistle as a Scottish national symbol was on a silver coin issued in 1470 by James lll.
CEILIDH (pron. Kay'-lee)
Roughly translated, this is a Gaelic term for "jam session". In my pocket version of Scots-Gaelic to English dictionary it's defined as "visit", but it can be further defined as a gathering of musicians, poets, storytellers, etc. and anyone else wishing to join in, sing along, or just listen and enjoy. After creating this knot I noticed what appeared to be a musical "G-clef" within the design, so the name just naturally followed.
This knot design went through several transformations before arriving where it is. I could have given it a couple of other names based upon its four distinct points - four elements, four seasons - but I chose compass because direction is something I think we all need now and then...and I can't say why but the design seemed to have a vaguely nautical feel to it as well.

As with "Ceilidh" this was named for an image that I saw within the knotwork - a horseshoe, which is commonly associated with luck. Hung over a doorway with the ends pointing up, the horseshoe will "catch" good luck and hold it in. Conversely, if it's hung with the ends pointing down the luck will fall out. Fortan is the Gaelic word for luck/fortune....and please make a note: it's just a name I gave the design because I had to call it something. No promises of good fortune come with the purchase of this Folk Wheel!

I originally designed this in black, gray & white for a wedding invitation. Then I thought it might also work well as a Folk Wheel, so I added some color and transformed it into a unique way to tell someone "I love you"!

All my original knot designs start out as doodles on graph paper. This one evolved gradually and was actually scrapped a few times until out of sheer frustration one day I scribbled in some leaves, and suddenly there it was. With a few strokes of my pencil it emerged from the scrap pile like a crocus from the long-cold earth, reborn....renewed. It's my hope that the design elicits a sense of life eternal for those who view it.

This design had no name or direction until it was nearly complete. It looked to me like some ethereal river with no beginning or end winding (or dancing, as it were...) through a star filled universe. No direct reference to the theatrical production intended, though Michael Flatley, Jean Butler, and company certainly were inspirational.

This is also one of my designs based on the triquetra or "trinity" knot, which lies in the center. The knot can be seen interlinked and repeated several times within the overall design.

Incorporated within this design are the five basic Celtic art forms: spirals, interlacing knotwork, key patterns, step patterns, and zoomorphics (plant and animal designs). It was created in tribute to Celtic art and the inspired brilliance of those original artists and artisans who gave it life.

Also sometimes referred to as the Druidic calendar, this design displays the eight Druidic feast days along with the trees, animals, elements, seasons, compass directions, and colors generally associated with them. These associations may vary from one Druidic sect or region to another, but to the best of my admittedly limited knowledge the most generally accepted ones are represented here.

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Basic Key Pattern
Basic Step Pattern
"Horse & Dragon" - Suzanne Shumaker
Although the cat doesn't figure quite as prominently in Celtic folklore as some other animals, it does have a place. Primarily they were thought to be guardians of the Underworld. Known as the Cat Sith (Scots gaelic) or Cat Sidhe (Irish gaelic) - both pronounced "kaht-shee" - they were believed to possess the power to steal souls of the recently deceased by simply passing over a corpse before loved ones had a chance to bury it. In the Scottish Highlands it was common practice to set a saucer of milk out on Samhein (Halloween) eve for the Cat Sith to drink, believing that out of gratitude it would bless the household for another year. Likewise, NOT doing so could result in causing all their cows' milk to dry up. In some instances a Cat Sith, also known as a Kellas Cat, was thought to be a witch in disguise - a transformation that could only be performed 8 times, for on the 9th attempt it would become permanent, causing the witch to remain a cat forever....quite likely where the notion about cats having nine lives originated.
Aside from their very real value as hunters and treasured companions, hounds also have a place in Celtic folklore and mythology. In Irish legend the hero Cuchulainn (aka "the Hound of Ulster" or "Hound of Culann") is said to have killed a blacksmith's hound in self defense. When Culann, the blacksmith, asked who would now guard his shop, Cuchulainn offered to take the dog's place himself, thus earning himself the title of "Hound of Culann". The offer was declined, however, and Cuchulainn went on to become one of the greatest warrior legends of that era. In Welsh mythology the ruler of Annwn (the Underworld) was said to lead a pack of supernatural hounds known as the Cwn Annwn, just one of several Celtic and European legends involving the "Wild Hunt" - a fey or ghostly group of huntsmen commonly associated with Woden. Seeing the Wild Hunt was believed to presage some catastrophe, such as war or plague or sometimes the untimely death of the individual witnessing it. In some instances it was also believed a person's spirit could be whisked away during his sleep to join the "Hunt". 

The wren was considered a symbol of wisdom and divinity, especially within the Druidic community. On the winter solstice a druid apprentice would go on a quest to find and capture one of the tiny and elusive creatures. The smallest and shiest were said to sing the loudest, but were also the hardest to capture. Should the apprentice manage to find and harness one of those it was believed he'd be blessed with insight for the coming year. This search represented the will power, courage, and perseverance required to awaken the Spiritual Self. It was also commonly believed that anyone who tried to kill a wren or steal its eggs would attract misfortune. Unfortunately, because of its association with pagan/druidic beliefs, it became a victim of suspicion and mockery upon the arrival of Christianity. As a means of driving out the old beliefs, groups of boys (aka "the Wren Boys") with blackened faces wearing straw masks and old clothes (often women's dresses) would go out on St.Stephen's Day, hunt and kill a wren (often by stoning it to death), then tie it to the top of a pole or holly bush and carry it from house to house, singing songs and demanding money. In return for payment the boys would give patrons one of the bird's feathers for good luck. This tradition still carries on in Ireland and several other places throughout Britain and Europe. However, thankfully for the wren, only fake birds are used nowadays.
As in many cultures throughout the world, numbers held special meanings to the Celts. The number 5, for example, represented the five directions: east, west, north, south, and center, as well as the five elements: earth, water, fire, air, and ether. The number three, however, was considered the most sacred. Dieties were frequently portrayed in groups of three, or triads, the most commonly known one being the Triple Goddess representing the three phases of womanhood - maiden, matron, and crone. It was also believed, among many other things, to symbolize the three "realms" - earth, sea, and sky...and the three cycles of life - birth, growth, and death/rebirth. 

A deeply held and sacred tenet of Celtic culture was the belief that life was a never-ending cycle of birth, growth, death/destruction followed by re-birth. The "Tree of Life" was a classic symbol of this, having its roots firmly planted in the underworld, its trunk resting upon the surface of the earth, and its branches reaching skyward to the heavens. In autumn its leaves would fall and eventually decay, becoming part of the earth again. Throughout winter its bare branch served as a reminder that life in the present is fleeting, but every spring the same branches burst forth with new life, imparting hope and assurance that death is only a temporary state and life, even if only in spirit, does go on.