Questions about variants of normal chess including bughouse, Chess960 (Fischer Random), atomic chess, racing kings, crazyhouse, antichess, etc.
A chess variation or chess variant is a game which is based on chess, but has different rules and/or layout and/or gameboard.
Some common chess variants include:
Chess960 (also known as Fischer Random Chess): the pieces on White's first rank are randomly placed (though the Bishops must be on opposite-colored squares and the King must be between the Rooks), and Blacks setup mirrors White's. The rules are the same as in standard chess.
Bughouse: a game with a partner with opposite colored pieces playing against another partnership. Pieces captured on your board are given to your partner, and he or she may place the pieces "in hand" on his or her board in lieu of moving a piece.
Displacement chess: a family of chess variants in which the rules are the same as in standard chess, but some of the pieces in the first rank of each player are transposed. Similar to Chess960, but not random.
Atomic chess: a capture results in an "explosion" which removes (aka. "kills") all pieces in any of the eight surrounding squares, except for pawns, including the captor. -Racing kings: The goal is to get your king to the other side of the board the first. Each side has all the major pieces, but no pawns, and they start on the 1st and 2nd rank.
Crazyhouse - pieces you capture from the opponent changes colour, and you can place them on the board. Similar to bughouse, but it is two-player, not four-player.
Antichess: the goal of antichess is to force your opponent to capture all of your own pieces. Similar to checkers, if there is a legal move that captures a piece, it must be made. Check is ignored - the king is captured like any other piece.
For a more comprehensive list of chess variants, see Wikipedia's list.