First and foremost Brythonic or Brittonic (1) was the language spoken by the inhabitants of Roman Britain. One of the five main groups of Celtic languages it developed differently from that other example of insular Celtic, the Goedelic under the influence of the Latin language of the Romans.
It is primarily distinguished from its Goedleic cousin by the difference in the pronunciation of the Indo-European sound "qu(kw)"; in the Goidelic this became represented by a hard "c" sound, but in the Brythonic it was replaced by a "p" sound. Hence the Goidelic for "son" becomes "mac", but in Brythonic it becomes "map" or "mab". And so Goidelic is known as Q-Celtic and Brythonic P-Celtic.
No complete sentence of Brythonic has survived, but the consensus is that it was cognate with Latin and that "its nouns had different forms in the nominative, the accusative and so on, together with the termination -os (comparable with the Latin -us)" (2). These terminations and forms disappeared as Brythonic mutated into its successor languages in the period 450-550.
Secondly Brythonic is the name given to the group of Celtic languages that were derived from and succeeded Brythonic itself, namely;
In the fifth and sixth centuries when Britain was invaded by a succession of Germanic or Anglo-Saxon warlords; the native rulers were forced back into the western enclaves of Strathclyde, Wales and Cornwall. Many also migrated southwards to Armorica, that is Brittany. Naturally this geographic separation meant that the original Brythonic spoken by the natives developed along different lines into four distinct but related languages.
Cambrian or Cumbric (which is the one people always forget about) was spoken in Strathclyde and died out in the tenth or eleventh centuries. Nothing of the Cumbric language has survived but it is generally considered to have been very close to Old Welsh. The last native Cornish speaker died in the nineteenth century although there are attempts being made to revive the language. Breton survives in the teeth of French indifference to European Directives regarding the protection of minority languages. Welsh of course, through a series of historical accidents is the most successful survivor.
Lastly Brythonic is used as a way of making a descriptive cultural statement about a group of people; so that Strathclyde for example, is a 'Brythonic' kingdom. It's a way of saying 'British' without confusing people and making a distinction between early medieval and modern concepts of what the word 'British' means.
(1) 'Brythonic' and 'Brittonic' mean the same thing. They are both derived from the Welsh word Brython for 'British', 'Brittonic' simply being an earlier attempt to anglicize the word. 'Brythonic' seems to be the more fashionable alternative at the moment.
(2) John Davies A History of Wales (Allen Lane, 1990)
(3) Sourced from the aforemntioned A History of Wales plus the BBC at www.bbc.co.uk/history/bytime/wales/brythonic.shtml