The Crabbe family is a most likely a pure-blood wizarding family, owing to their close ties to Lord Voldemort. Like many who supported the Dark Lord, they had prejudices against Muggles, Muggle-borns, and sympathizers. Notable members include Vincent Crabbe and Crabbe Sr. who was a Death Eater.
Vincent Crabbe's ancestry is never mentioned in the books, however, in 2001 Rowling flashed a notebook during a BBC interview that showed her notes on the students in Harry Potter's year and Vincent is noted as being a Pureblood. This information cannot be considered canon, however, because the notes conflict in too many places with the stories as they were actually published.
The Crabbe family is connected to the Black family because Irma (Crabbe) Black (Vincent Crabbe's mother) married Pollux Black.
“Krabben” Low Ger. ‘to scratch, claw.’ To be crabby is to be bad-tempered and combative, or to complain irritably (OED).