The act of chugging too much beer and as a result vomiting - then catching said vomit in your own hand. "
"Bilge" refers to the bilge on a ship. The bilge is where all disgusting things on the ship are kept (I.E. Used oil)
In the Boston Bilge, your hand is the bilge.
Guy 1: Dude, did you see that guy Chris last night?
Guy 2: No, what happened?
Guy 1: He had this girl all over him up until the last game of beer pong.
Guy 2: What happened?
Guy 1: Boston Bilge
Guy 2: HAHAHAHAHAHAHA. What a tool. Let us call and mock him.