In short it means "tribe" or "people" descending from Brian.
So why is "Brian" (an Irish name) spelled so strangely?
Well, in Irish, as in other languages, there is what is known as the "genative" case, that being a case of a word to denote possession.
Brian here must be placed in the genative case. In Irish that is done by "aspirating" the first letter of the name as indicated by placing an "h" after the "B" to become "Bh". Which also mutates from a "b" sound to a "v" sound in this case.
However, we are not done here. In the genative case we must also "slenderize" the last letter in the word as well. That means we must place either an "i" or an "e" (the two slender vowels - the rest are considered broad) before the final "n" in Brian
a slender "n" is pronounced more shortened like the "n" in "finger" as oppsed to a broad "n" like in "fun"
therefore:
Brian (bree - uhn)
becomes
Bhriain (vree - in)
Thats the quick and dirty explanation anyway.