No problem, dude! After all, everybody that's here is here to yak!

A "button" is pretty much synonymous with a "made man," "wise guy," or "enforcer." My Mafia comrades will correct me if I'm wrong, but all these labels are pretty much used interchangably (at least in the Mafia stuff I've read). It's someone who's "made his bones" (i.e., killed someone who the Boss ordered whacked), and thus became a "button." "The Boss tells me to push a button on a guy, I push a button," so said Willie Cicci in GF 2. Technically, I suppose one isn't considered a "made man" (i.e., an official member of a Family) until he's undergone the ceremonial initiation rites. Usually, this involves cutting the finger and allowing it to bleed over a candle and a "prayer card" of some saint. As the candidate stands there, the Boss or the candidate's sponsor recites the Cosa Nostra rules, e.g., never covet another made man's wife, never be a rat, etc. After the candidate has agreed, the saint's card is burned in the candle, the idea being that the same will happen to the candidate should he ever break this oath he's taken. Okay, now my fellow Mafia experts can correct me if I'm wrong or if I missed anything. How'd I do, comrades?
