I think bughouse is the most popular of chess variants (if you don't count national variants like Xiangi or Shogi). It is usually played by 4 people, but I think 2-player version should be fun as well.
Chessmaster1000: Yes, you are right. It is not so easy for white to win. May be the game is a draw. I don't see how white can continue attack without bringing own King into danger.
Fencer: Hm, looks like white has a forced win with this addtional rule: 1. Nb8-c6 Ng1-f3 2. Nc6-b4 (threatens 3. Nb4-d3++) Nf3-e5 3. Ng8-f6 N b1-c3 4. Nf6-g4! and white wins.
Nightstorm: Probably it is better to play Shatranj as Chaturanga rules are not exactly known. Some historicans even doubt at all if chaturanga really existed.
I played several games of Shatranj against ChessV and it is quite interesting. The only problem is that games are usually very long, probably twice as long as in chess.
I found this interesting site: Экзотические шашки (Exotic checkers). It is in russian language, but there are enough diagramms. Beside other checkers variants, it contains three chess+checkers variants.
Yes Kevin, you win by either checkmating your opponent's king, or disallowing your opponent to have his anti-king threatened. Anti-King cannot be captured.
More detailed rules can be found on http://www.chessvariants.com/diffobjective.dir/anti-king-chess.html, see "Anti-King Chess II".
There is one great chess variant, which is little known, but is very interesting: Anti-King Chess (invented by Peter Aronson). All rules are the same as in standard chess, except that a new piece is added: Anti-King. The Anti-King is under check when it is not attacked! It is a checkmate, if it can't leave such position. The Anti-King can't capture enemy pieces. At the start white Anti-King is placed on d6, black on d3.
I played this chess variants several times against Zillions-of-Games, but Zillions is not especially strong in it. It would be much more fun to play this chess variant on BrainKing!