Celts, Gauls, Galatians - different names, used in different places and at different times, but they all mean the same - a population of peoples who rose somewhere in Central Europe during the Neolithic and then dispersed in all directions, mixing with local populations, creating separate groups and ending up as a mystery for most of the world. Some of these names indicate a single group, some of them are more generic but a lot of ancient authors used them interchangeably at one time or another and there are archeological and other evidence to support the connection so we know that group existed. But who they were? And why Central Europe when everyone knows that these are Western European (and British Islands) people? Well... about that... popular culture and actual history have a different opinion on these.
There probably was no better author for this book than Cunliffe - while not everyone agrees with him on every topic, he is an archaeologist who spent spent his life excavating the English countryside (apparently there is more to it than murders and mayhem as the crime authors will make you believe) and writing extensively on Iron Age Britain and Europe. Which is where the Celts come into the picture - they match his period.
The story he tells has multiple distinctive parts - from the first remains of the peoples, to the development of what will be known as Proto-Celtic language (that's one of the topics where a lot of the current scientists disagree with him - where did the language arise actually: the book explains his theory on the topic although it does mention that there are other theories - but then what can you do in a book of 145 pages), the dispersal of the tribes which made up the initial centers and their mixing with the locals they found elsewhere (all of that seen mainly through pottery - pre-history and early history deals with a lot of pots). But then as time progresses, the authors of the classic period start mentioning them (well, not always in a very nice light) and more and more artifacts start pointing to the history and how it goes (they even made it into the Bible as the Galatians, following one of the known dispersal waves to the East).
And they kept moving and mixing; somehow managing to keep their legends and a language group alive (and somewhat well with four continuously living languages (Breton, Irish, Scottish Gaelic and Welsh)) - and the linguists can say a lot about these movements because of how the languages developed and are used today. And that's where it gets a bit convoluted because the Celts of these ancient times had mixed so thoroughly with everyone that these days the Celts are defined as the speakers of the Celtic languages -- which makes the question of where these languages developed and how they dispersed a very hot topic indeed.
Once Cunliffe is done with the pre-history and the Roman empire, things get even more complicated in his narrative because more and more people are moving across Europe (and the British islands), displacing populations, mixing with whoever they find and pushing the old inhabitants (some of which were once the new ones) to the corners. Probably one of the most ironic fact around the whole situation in the middle ages is that it was Julius Caesar's "The Gallic Wars", republished in 1469 and made public in Venice in 1511, that reminded everyone of the whole Celts/Gauls situation, which led to histories that were not rooted into the Bible and the emerging states of Western Europe started using that history to write their own histories - and the ancient Celts and Gauls showed up back on the map - reinvented and adding the stamp if secular antiquity into the story of the new nations. Cunliffe gives an overview of how that developed in the 16th century and beyond, leading to the current Celtic mania - from books in Latin early on through the universal histories which followed them to the first festivals and congresses of and for Celtic culture to the early 21st century where "Celtic" art and culture is everywhere - not always meaning the same for everyone and not always connected to the historical roots of the peoples.
So who were the Celts? That really depends on why you want to know that - the answer may surprise you anyway you look at it.
The one thing I really disliked about the book is that Cunliffe forgot that he was writing a short book. So every few (short) chapters, he will have an "interlude/review" chapter which added almost nothing new (it did some synthesis but... as dense as the text is, there is just not enough material to require that). But that is a minor gripe.
As usual, there is a "Further Reading" section, which is heavily curated to include mainly works in English with lengthy bibliographies (and skipping the "lunatic fringe").