Bridge

Bridge: The most convoluted game you don't play.

Short version? It's a trick taking game involving bidding. If you don't make your bid, you lose points.

First complication? While you only score if you make your bid, the points you make for going over your bid are usually not worth it compared to your actual bid, and thus you have a heavy incentive to get your bid exactly correct.

Second complication? You're playing with a partner who can't see your cards, and the only method you are allowed for communication is your bidding.

Third complication? The defenders know what cards one of the bidders has (although, then, so does the bidder), so they can partially control the game that way.

This leads into the fourth complication: People trying to send messages about their hands with their bids to optimize their bidding. If you've ever seen Bridge columns in newspapers, they're usually all about bidding.

Fifth complication? If your opponents think they can beat you during the bidding, they can "double" your bid, making the deal worth more points. You can in turn "redouble" and make it worth even more points.

Other then that, it's a fairly standard trick taking game.

Duplicate Bridge

Bridge players, fools that they are, frequently want to know how they compare to each other in skill in tournaments. Since, in theory, each hand is shuffled at random, this poses a problem. The usual solution is "Duplicate Bridge", where everyone's hand at a given table is fixed ahead of time by the tournament organizers, and partners move between pre-set-up tables playing these fixed hands, with things going by total score.

Of course, somebody occasionally accidentally fucks up and changes the cards around, leading to lots of fun when attempting to figure out who "won".