For a long time, the argument of 'invasions' lost currency in the world of archaeology in favour of a 'pots not people' theory when it came to the Celts and other migrations. It was argued that there was no Celtic invasion, and indeed there is no substantial evidence of a large ingress of people to Britain during the "classic" Celtic (Halstatt / La Tene) era. The notable exception is the invasion of southern England by the Belgae tribe of the Low Countries, as attested in Caesar's Commentaries of the Gallic Wars.

Recent DNA discoveries have given credence to another, and to my mind more believable, theory - that of large-scale replacement of the population prior to the rise of Celtic culture in central Europe.

This idea states that the haplogroup R1b arose in the Russian / Ukrainian Steppe in the late Neolithic, and is associated with a group of Indo-European pastoralists who had domesticated the horse and used wheeled transport. They swiftly overran Europe, invading and subjugating to the Western coasts in early Bronze Age. In Western Europe they may well be associated with the Beaker Culture. Note that this all happened over 1,000 years before the earliest possible rise of central European "Celtic" culture.

So the theory is that the British, and Irish, people, have very much in common with the Bronze Age inhabitants. And although there was much immigration into the Isles from the Continent and Scandinavia, in addition to two-way migration between Britain and Ireland, this did not leave as much genetic imprint behind as did the Early Bronze Age invasions.

This theory also posits that the Insular Celtic languages - Gaelic, Welsh, Cornish, Breton etc - have their roots in the Bronze Age, and are not some cultural lingua franca that was adopted by the indigenous inhabitants of the islands.

This is a theory that is growing legs, and I'd be very interested to hear the opinions of others on this whole area.

Sources:

Celtic from the West, Cunliffe & Koch

Origins of the Irish, Mallory

The Isles, Davies

Ancestral Journeys, Manco

To expand a bit on what has been said, the ancient people of Britain and Ireland were never really called 'Celts' by an ancient source. That association only came about at the turn of the 18th century after it has been established that the Pre-Germanic languages spoken in the islands like Irish and Welsh were related to that of the ancient Gauls, who the Romans also termed 'Celts'. There is a growing consensus, specially amongst archaeologists of Britain and Ireland that the word 'Celtic' is problematic because it subscribes to and perpetuates erroneous ideas which equate language and art style as markers of ethnicity or 'nationhood'. In the past two decades, you'll see the word 'Celtic' being increasingly avoided or in inverted commas when used at all.

There is enough on this sub alone to expand on why this term is now problematic. Besides /u/innisfallen's answer above, you might want to look at this recent thread where /u/krazyhades compiled answers by some other users. I'd also like to suggest a chapter called 'Celts?' in Jones's book cited bellow which I think does a great job at explaining everything to a lay audience. One of the points he makes that I really like is how he compares the way Romans used the word to the way European colonisers called members of all the contrasting societies they encountered in the Americas 'Indians'.

While it has fallen out of favour amongst archaeologists, it still remains a linguistic term, that is, Celtic, nowadays refers to a language family rather than an ethnic group or culture. This said, some archaeologists still propose Celtic can be used to refer very broadly to the cultures of Pre-Roman Europe that shared some characteristics with one another. Think about how our modern concept of 'Western culture' doesn't really refer to a single culture, let alone ethnicity, that shares one single origin rather than different cultural trends that seem to cluster together only when compared to other groups.

However, that only partially answers your question. Fair enough, if we can't really talk about 'the Celts' how come the peoples living in Britain and Ireland spoke Celtic languages related to those spoken by other groups that were also lumped together under this label? Celtic is a branch of the Indo-European languages, which most probably arrived to Europe in the Chalcolithic ('Copper Age') with people who moved from the Pontic Steppes located in what is now roughly Ukraine and southwestern Russia. Given that extracting (enough) DNA from ancient bodies has only become possible recently, studies about this topic have only been published within the last decade. At first, they seem to point at a rapid 'invasion' bringing back the idea of massive migrations as the vectors of cultural (and linguistic) change that were considered outdated amongst archaeologists. However, some scholars were quick to point out that what migrations aren't single genocidal event, and that genetic evidence can also be the result of minor intrusions or continuous contact which after a few generations can significantly alter the genetic makeup of a population.

This is relevant to our discussion because in the British Isles there is very little evidence for the movement of peoples into the region in the Iron Age. The examples of what gets generally termed as 'Celtic art' in Britain and Ireland indicate they are either imports or local adaptations to what was fashionable at the time. Assuming you live in Britain yourself, think about how the fact you might have IKEA furniture in your living room doesn't mean there was a significant migration from Sweden, let alone that the appearance of IKEA furniture and Absolut bottles in the material record marks when Germanic started being spoken in Britain.

Where there is enough evidence for the possibility for a significant movement of people, however, is the Bronze Age. Recent ancient DNA studies also suggest a significant movement of peoples into the islands in the Bronze Age who were genetically related to the proposed first speakers of Indo-European mentioned above. This has already been a hypothesis held by many archaeologists for a good time, but has recently culminated with the (controversial) ideas of archaeologist Barry Cunliffe and linguist John Koch in their 'Celtic from the West' volumes in which they propose that once Indo-European had arrived to the Atlantic Fringe, it develops into Proto-Celtic, which became a 'lingua franca' used in trade in the region and eventually spread into Central Europe. The most controversial parts of their work is that they identify NW Iberian scripts as the earliest evidence for Celtic and associate the spread of Celtic with the Beaker 'phenomenon' that also originated in Iberia and is associated with the spread of metallurgy in Atlantic Europe. Thing is, depending on whom you ask, you might hear the Beaker phenomenon was anything from a group of people, an ideology or just a fashion.

This also seems more linguistically sound because while Celtic is a still poorly understood branch of Indo-European mostly due to the lack of evidence, it has long been argued that the branches are too diverse to have been the result of a single expansion beginning in the Iron Age. The classification most people are familiar with, which divided the Celtic branch into a 'Q-Celtic' and a 'P-Celtic' family is now obsolete and most scholars talk about 'Insular' and 'Continental Celtic' instead. However, the fact languages spoken in Ireland and Iberia share some conservative features while the languages spoken in Gaul and Britain share some innovations also seems consistent with the fact that what we see in the archaeological record in the Iron Age is that Ireland became increasingly isolated while Iberia started turning their attention to trade with the Mediterranean, whilst contact between Britain and the continent north of the Pyrenees remained constant.

It would be an exaggeration on my part to argue for a consensus. Given how 'the Celts' play at least some part in nationalistic ambitions of basically every Western European country or group, discussions tend to get passionate and somewhat biased very quickly. However, I think the tendency, at least amongst Anglophone archaeologists, is to increasingly argue the Celtic languages were a result of acculturation that happened in the Bronze Age mostly amongst peoples who would already have spoken some form of Indo-European and didn't rely on migrations but on the exchange of goods, technology and ideas.

Sources:

Anthony, D.W., 2010. The horse, the wheel, and language: how Bronze-Age riders from the Eurasian steppes shaped the modern world. Princeton University Press. Cassidy, L.M., Martiniano, R., Murphy, E.M., Teasdale, M.D., Mallory, J., Hartwell, B. and Bradley, D.G., 2016. 'Neolithic and Bronze Age migration to Ireland and establishment of the insular Atlantic genome'. Proceedings of the National Academy of Sciences, 113(2), pp.368-373.

Collis, J., 2003. The Celts: origins, myths & inventions. Tempus Pub Ltd.

Cunliffe, B.W., 2001. Facing the ocean: the Atlantic and its peoples, 8000 BC-AD 1500. Oxford University Press.

Cunliffe, B.W. and Koch J.T., 2010. Celtic from the West. Oxbow Books.

Cunliffe, B.W. and Koch, J.T., 2013. Celtic from the West 2: rethinking the Bronze Age and the arrival of Indo-European in Atlantic Europe. Oxbow Books.

Cunliffe, B.W. and Koch, J.T., 2016. Celtic from the West 3: Atlantic Europe in the Metal Ages : questions of shared language. Oxbow Books.

Haak, W., Lazaridis, I., Patterson, N., Rohland, N., Mallick, S., Llamas, B., Brandt, G., Nordenfelt, S., Harney, E., Stewardson, K. and Fu, Q., 2015. 'Massive migration from the steppe was a source for Indo-European languages in Europe'. Nature, 522(7555), pp.207-211.

Jones, C., 2004. The Burren and the Aran Islands: exploring the archaeology. Collins Press.

James, S., 1999. The Atlantic Celts: ancient people or modern invention? British Museum Press.

Manco, J., 2013. Ancestral Journeys: The Peopling of Europe from the First Venturers to the Vikings. Thames & Hudson.

Olalde, I., Brace, S., Allentoft, M.E., Armit, I., Kristiansen, K., Rohland, N., Mallick, S., Booth, T., Szécsényi-Nagy, A., Mittnik, A. and Altena, E., 2017. 'The Beaker phenomenon and the genomic transformation of northwest Europ'e. bioRxiv, pp.1-28.

Waddell, J., 1995. ‘Celts, Celticisation and the Irish Bronze Age’. Ireland in the Bronze Age, pp.158-69.

Check out /u/Miles_Sine_Castrum's response to What caused the Celtic expansions across Europe, and then the later Germanic expansions? Were there any shared causes? and /u/Daeres' response to Who were the ancient "Celts" that lived throughout Europe during the time of the Roman Empire? Why is it today anything "Celtic" usually always refers to only Ireland?.

As a follow up. Do we have any surviving examples of Celtic ships? Like how some Viking era boats have been recovered?