*based upon my own limited research...
corrections and/or additional information welcome!
It should be noted  that no clearly definitive, uniformly accepted account exists of how the Celtic culture and language came to the lands now known as the five Celtic Nations of Great Britain (Ireland, Scotland, Wales, Isle of Man, and Cornwall). Whether it was by invasion or passive migration is not clear, though recent studies lean toward migration and a gradual absorption and diffusion of Celtic language and customs into the pre-existing, non-Celtic population. What language those earlier Neolithic people spoke is not known. The earliest traces of Celtic culture in Britain seem to place their arrival there sometime around 500 BC.

From an anthropological perspective it is primarily language and, to a slightly lesser degree, art/artifacts, mythology, and societal customs that define what is (and is NOT) historically Celtic. The term "Celtic" does not, as some might believe, describe a particular "race" of people with common ethnic backgrounds or genetic markers. Celtic ancestry is complex, diverse, and far more difficult to trace. Recent genetic studies have shown that "the vast majority of Britons have ancestors from the Iberian Peninsula, as a result of a series of migrations that took place during the Mesolithic and, to a lesser degree, Neolithic eras"....in other words, long before the arrival of any Celtic speaking peoples. Author and historian Bryan Sykes  ("The Blood of the Isles") states that the genetic structure of Britain and Ireland is "Celtic, if by that we mean descent from people who were here before the Romans and who spoke a Celtic language." He further speculates that the vast majority of the inhabitants of the British Isles, whether they consider themselves "Celtic", "Anglo-Saxon", or otherwise, are descended from the original Mesolithic hunter-gatherers who migrated north from Iberia at the end of the last ice age, or around 13,000 years ago.
Archeologically speaking, the beginnings of Celtic society can be traced back to Iron Age Europe. The word "Celt" itself is derived from "Keltoi" - a term used by ancient Greeks to describe all who lived north of the Alps. The earliest significant traces of Celtic culture discovered thus far lie in the Hallstatt region of Austria and date around 800 BC, or the very beginning of the Iron Age. Artifacts uncovered there in the early 1800's revealed the existence of an advanced society of iron workers, farmers, and traders with a unique set of cultural markers now defined as Celtic. In 1845 another trove of artifacts was discovered in the small village of La Tene, Switzerland, similar in nature but more elaborate in design and opulance. These have been dated to around 500 BC. These two archeological discoveries are now considered to be the defining markers for early Celtic societal development. Art and artifacts associated with ancient Celtic culture are commonly assigned one of these two time periods, either Hallstatt or La Tene, based upon form, development, and degree of sophistication (not necessarily upon place of origin).
From around 800 - 450 BC Celtic speaking people occupied most of central Europe. Expansion into the Iberian Peninsula (now Spain and Portugal) and the British Isles likely occurred during the late Iron Age, somewhere between 500 - 1st century BCE. These early Celts were a tribal society. They did not exist as a single nation, but as many separate states or clans ruled by kings or chiefs. Customs, artwork, and belief systems often varied from tribe to tribe, though archeological evidence has shown that some similarities existed as well. What is important to understand is that they were a widely diverse people bound by a common language and, to some degree, customs and beliefs - not by any shared ethnicity or centralized government. The structure of Celtic society as a whole appears to have been divided into three groups: (1) a warrior aristocracy, (2) a class which included professions such as druids and bards, and (3) everyone else. Little else is known, which leads us to the main problem surrounding Celtic historical research - they didn't write anything down! Instead, customs and history were passed from one generation to the next through a strong oral tradition. Most of what we've learned is based upon archeological evidence and comparisons with other co-existing cultures...which leaves a great deal to speculation. The only written accounts of Celtic life that exist today were composed by outsiders, such as the Roman historians Livy and Diodorus, and we have to consider the possibility that such accounts may be tainted by perceptions that were not altogether objective.
It seems unfortunate that much of how the ancient Celts lived and loved, struggled and died will forever remain a mystery, but their legacy lives on in the rich languages, music, dance, and art that have managed to survive. Despite the inexorable progress of time and technology, these basic elements of a unique and colorful culture have managed to endure, lending strength to the hope that they will continue to thrive for many more centuries to come.
Sources: www.britainexpress.com/History/Celtic_Britain.htm
What does it mean to be Celtic? The term is often thought to be interchangeable with all things Irish, and of course Ireland has long been and remains to be a great stronghold of Celtic culture. But it's not the only place where it exists. There are actually seven "Celtic Nations": Ireland, Scotland, Wales, Isle of Man, Cornwall, Brittany (western France), and Galicia (northern Spain). These geographic areas were determined according to the degree of Celtic language and culture that continue to exist and grow there.
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Neolithic stone carvings - Newgrange, Ireland
Artifacts from the Hallstatt period (c800 - 450 BCE
Golden neck ring -
Hallstatt culture, 550 BC 
Sun disc brooch - gold plated bronze, 4th century BC   La Tene period 
Discovered in France
Detail from Gundestrap cauldron, 200-100 BCE   La Tene period
Discovered in Danish peat bog, 1891