Combinatorial game theory studies games in which two players take turns, each player has complete information, and there is no chance involved. A game is over when there are no legal moves left, and typically the winner is determined by who made the last move. In normal play, the last player to move is the winner, and in misère play, the last player to move is the loser.
Misère play has a much more complicated theory than normal play.
A game is impartial if it satisfies the additional property that the legal moves from each position are the same for both players. (This excludes, for example, chess, in which one player plays the white pieces and the other plays the black pieces.)
These pages contain a few random things on impartial games.