A term given to a person (typically a male) who acts or does something in a "hard-ass", asshole, and/or tough-guy manner. The Billy Badass themselves, of course, do not think they are a "Billy Badass"; the term is usually told to, or directed at, someone in order to poke fun at, or to express contempt for, their prior, recent, or planned excessive or flamboyant actions.
Guy 1: "If this cop gives me any shit, I'm gonna smash his face in".
Guy 2: "Yeah, whatever...Billy Badass".
A little Punk teenager comes strutting into a store like he owns the place.
Store Clerk: "Check out this Billy Badass".