XiangQi is a contender for the most enjoyable two player "chess" game of them all.
The moves of the Chariot, Horse and Soldier are similar to their Western counterparts, Rook, Knight, and Pawn respectively, but the Elephant, Guard and (especially) the Cannon are quite different. Furthermore, restrictions placed on certain pieces by the presence of the dividing "river" and the two "fortresses" on the board make XiangQi a unique and exotic experience.
Fewer soldiers (pawns), coupled with the unique attacking capability of the cannons, means the action starts from the very first move. This may feel strange to those who are more comfortable with the slow, measured build up of Western Chess.
The Chinese characters that differentiate the pieces may take a short while to familiarise yourself with but, as with Mah-Jong, this adds to the picturesque nature of the game.
Cheap plastic sets with roll-out paper boards for a few dollars/pounds, up to beautifully carved wooden sets and boards should be available from your local Oriental emporium. Just ask for XiangQi (pronounced "shyang-chee").