Chess 960
Chess960 was created by World Champion Bobby Fischer. The base game is played entirely the same, except the back row of pieces are changed. The position of the knights, rooks, bishops, queen, and king should be completely random as long as both sides start the same way. It's named 960 because Fischer determined there are 960 different possible starting combinations. The Chess960.net site says the game still preserves the fundamental aspects of chess, but it renders useless any guides that focus squarely on chess opening theories.
Peasant's Revolt
Peasant's Revolt players start with different and unequal pieces. The White player represents the peasants, and has a king and a full row of pawns called the "mob." The Black player controls a king, a single pawn and three knights. Black has a heavy advantage early, but if White can get pawns to Black's home row, those pawns can become any other piece. White could end up with his own army of rooks, bishops or queens.
Anti-Chess
Anti-Chess follows regular chess rules, except players are required to take any piece of which they are able. The goal is to lose all pieces (except the king), but a player can still win by checkmate. So a player who has lost most of his pieces should strive to lose the rest, while the player with the most pieces goes for a checkmate.
Xiangqi (Elephant Game)
Xiangqi, or Chinese chess, is played on a board divided into two halves by a "river." In the center of the back rows is the palace, a 3 by 3 square that the king and his two guards can never leave. The "horse" replaces the knight, and it cannot leap over other pieces. The elephant is closest to the bishop, and can move two (and only two) spaces in a straight diagonal line. The canon has no direct equivalent in Western chess. It normally moves like a rook, but can only capture pieces by leaping over a middle piece of either color.