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The player whose opponent resigns always wins the game; check FIDE Laws of Chess. (I also thought that it might be a draw if the opponent can't checkmate, but turns out that nope, the opponent wins.)
Going off topic from the forum title, seems like this also means that if I resign when I have a king and a queen and the opponent only has a bare king, with the position not being in checkmate or stalemate, my opponent would have won by my resignation.
Yup, Eric. The only time that resignation doesn't end the game is when there's checkmate or stalemate on the board. It's hard to imagine playing checkmate, then resigning though.
Rightfully, checkmate takes precedence over resignation, so even if the person playing the move is unaware that the move that he just played happens to be a checkmate and opts to resign, the correct result is still a checkmate.