Did I get this right? Please, feel free to make corrections; I'm actually genuinely interested if I failed or succeeded. I see people arguing about this all the time.
Celts - broadly speaking, were the people living in the British Isles at the time of the Roman conquest. They spoke languages that were the ancestors of modern Breton, Welsh, Scots Gaelic and Irish.
Saxons - (and their neighbours the Angles and Jutes) started arriving in the British Isles from about the 5th century AD. They came from what is now Denmark, North Germany and Friesland, though they probably lived further East before that. They spoke Germanic languages. These languages (usually lumped together as "Old English" or "Anglo-Saxon") became dominant in most of England and southern Scotland, while Cornwall, Ireland, Wales and northern Scotland remained largely Celtic-speaking.
Gallic is a tricky word: the Romans called most of north-western Europe "Gallia", so it gets used in several different ways, generally to refer to people or languages of Celtic origin. Nowadays, we tend to use "Gaelic" (pr. "gallic") for the Celtic languages of Scotland and Ireland; and "Gauls" for the Celtic people living in France during Roman times.
Franks - were Germanic people who became dominant in most of NW Europe around the 6th century. They originally spoke a Germanic language, but those living in what is now France later became French-speaking. Clovis and Charlemagne were Franks. Confusingly, Muslims in the medieval period tended to refer to all Western Christians as Franks.
Normans - were from Normandy, in northern France. They seem to have been a mixture of local people (Franks/Gauls/whatever) with Vikings who came there from Norway around the ninth century. They spoke a dialect of French (a Romance language). Normans later spread to England (1066!), South Wales, Ireland, Sicily, and quite a few other places, taking their language with them.
