I'm new at chess and still learning the rules. The other day I was playing a friend of mine, and it got to the point where I had only my king and queen and he had only his king and bishop. I forgot the exactly how we got to this point, but when we did, he said it would be stalemate and result in a draw. I argued that we keep playing and I wound up putting him in checkmate. He didn't make any fatal or illegal moves; I just beat him. My question is, why would this be a draw if I could win? Everything I find says that when either side has only a king and bishop/knight it results in a draw. Of course I may have misunderstood, but it seems like one side can get punished for simply taking out the other sides pieces.
If I understand well you had a king and a queen, and your opponent a king and a bishop.
It is not a stalemate nor a draw, unless you reach a position like this one, for example:
[fen "8/8/8/8/8/6K1/8/5Qbk b - - 0 1"]
A stalemate occurs whenever the player having the turn has no legal moves and he's not under check.
A draw happens when there's no sequence of legal moves (even if they are absurd ones) than can lead to a checkmate, when the same position is repeated at least three times, or when at least 50 moves have been done without a pawn move or a capture.