The Celts of Gaul (roughly modern day France) settled in Britain in the centuries after 600 BC. By the time that Gaul and Britain were brought into the Roman Empire (most of Britain was conquered by 85 AD) these lands shared a language which linguists call Gallo-Brittonic. This was an Indo-European language, just one of nine different branches.
From Gallo Brittonic descended Welsh, Cornish and Breton, the so-called P Celtic languages. (The Indo-European kw sound had developed into p). The Celtic spoken in Ireland and the Isle of Man became known as Q Celtic (the kw sound had been retained and was written first as q and later as c).
The Brittonic language survived Roman rule in Britain but the Anglo Saxon invasions which followed the departure of the Roman garrisons around 400 AD led to the language being largely supplanted by Old English.
Wales is the only part of Britain where a version of the old Brittonic language has survived. The language developed over the ages becoming Welsh in the period between 400 and 700 AD. It became threatened after the Norman conquest in 1066 when Anglo-Norman incursions pushed back the language frontier. The 19th century industrialisation of South Wales and associated immigration from other areas of Britain led to a further decline but these days there is a feeling that the tide has turned. Recent years have seen the establishment of new Welsh medium schools and an increasing awareness that Welsh represents more than just a means of communication.