The most common odds I've played (a piece often too much) is pawn and move. This usually means that the better player has Black and his f pawn removed.
Additionally, when playing piece/rook/queen odds, the odds giver plays white and usually has the moves a3 and h3 already played. The idea of this is to prevent the odds receiver playing for simplification to a won endgame, they are forced to play a whole game.
One player has more time than the other.