Different types of chess (that still use the same pieces and board):
Tournament chess - play using chess clocks, when your time is up, you lose. There are different settings for play times, based on the level of competition. At international level, the official rules are 2 hours for 40 moves.
Lightning chess is a variant using short time settings per game. Some grandmasters consider less than 60 minutes per game per player to be lightning chess. Personally, I feel that 5 minutes per game per player is the most fun.
Suicide chess - the first player to lose all his/her pieces wins. During each players turn, any piece that can be taken must be taken. The King acts as just a piece with a one square move in any direction rather than as the piece you must protect at all costs.
Transfer chess - a really fun variant where two players take on two other players. Two boards are set up side by side with colours alternating (each team has one player playing White and one player playing Black). The fun begins whenever there are captured pieces. Any of your opponent's pieces that you capture, you hand it over to your partner who may then, during his turn, place on any empty space on the board (some players disallow placing pieces on the eighth row). This rule goes for all players, so pieces get recycled into play very quickly on the other board. The game ends whenever any player's King is checkmated, when the opposing team is declared winners. Note: pieces can be placed to block, so checkmates aren't always as easy as they seem for the uniniated. Transfer chess usually requires the use of chess clocks to prevent players waiting incessantly for pieces to place. Lightning transfer chess is really fun.
/dev/joe has informed me that transfer chess is also known as bughouse
Something a little less conventional perhaps, is Omega chess which requires a whole new board (10x10 with extra 'wizard' spaces) and two extra piece types - the Wizard and the Champion. Read about it at: http://www.omegachess.com/
There is also minister chess, which is played on a 9x9 board with two Queens but the Omega chess stuff sounds sexier.
Update (July 4th, 2000): I've noitced that, in Shanoyu's writeup, he's forgotten to mention that perpetual check is another way to force a draw.