I have a couple of drinks worth of experience with a number of entertaining variants, which have no official name that I'm aware of: (EDIT: The first is "progressive chess" and there are a few different rule sets for handling checks: http://en.wikipedia.org/wiki/Progressive_chess)
- Accelerating chess. White gets 1 move, Black gets 2 moves, White gets 3 moves, and so on, until someone wins. Rule to make it not suck: You can't put someone in check during your intermediary moves. This involves more skill than you might expect, and if your opponent is an adept defender at the end of his turn(s), it can be surprisingly hard to mate without putting the enemy in check, even with 5, 6, or 7 moves at your disposal. Sometimes you will find an extremely clever combination and feel very special.
- "Camelot." You can jump your knight on top of your rook, at which point it can move like either a knight or a rook. If you like, you can jump it back off afterward. If someone captures it, they capture both. Bonus carnage rule: You can move it like a rook AND proceed to jump the knight off in the same turn. This leads to hilariously powerful and unbelievably bizarre mate threats.
- Additional rule that works well with Camelot to add variety: X-ray bishops. Your bishop can move through enemy pieces and capture a piece which would normally be concealed by a pin. I bet you think this is a dumb ability but try it out. You have to be careful not to get mated on move 2.
When I was in college my friends and I came up with a chess variant that used the dice from Risk. You played chess normally but if you didn't like the move that your opponent made you could 'challenge' it using the dice. Since you were challenging a legitimate move you would roll the two red dice and the defender would roll the three white ones. Whoever had the higher total would win. This meant that if you put the other player in check you still had to actually capture the king because they could challenge the actual capture and keep the game going.
I think that if you lost the roll not only did you have to take your move back but you lost your turn as well. I remember it favored a style of play in which you could take your queen and start attacking pieces on the opponents back row. If they tried to capture it you could get lucky on the roll and keep attacking.
This reminds me a little of the combinatorial game theory game done on Go endgames: fascinating but impractical!
There are basically three major chesses; western, Xiangqi and Shogi, and three popularish Western variants (Chess960, suicide and bughouse). All of them are decent games.
(fwiw, ~2000 ELO at chess, ~3 kyu at Go. competent but unspectacular.)
Amusingly, that page says it was last updated November 1996. It still renders decently today, probably due to eschewing anything but the barest of HTML.
There's yet another "Last modified" date with a year of 2002. I think that over its long history, the page has been through a few different layers of CMS.
- Accelerating chess. White gets 1 move, Black gets 2 moves, White gets 3 moves, and so on, until someone wins. Rule to make it not suck: You can't put someone in check during your intermediary moves. This involves more skill than you might expect, and if your opponent is an adept defender at the end of his turn(s), it can be surprisingly hard to mate without putting the enemy in check, even with 5, 6, or 7 moves at your disposal. Sometimes you will find an extremely clever combination and feel very special.
- "Camelot." You can jump your knight on top of your rook, at which point it can move like either a knight or a rook. If you like, you can jump it back off afterward. If someone captures it, they capture both. Bonus carnage rule: You can move it like a rook AND proceed to jump the knight off in the same turn. This leads to hilariously powerful and unbelievably bizarre mate threats.
- Additional rule that works well with Camelot to add variety: X-ray bishops. Your bishop can move through enemy pieces and capture a piece which would normally be concealed by a pin. I bet you think this is a dumb ability but try it out. You have to be careful not to get mated on move 2.