Badminton is a sport that is somewhat similar to tennis, in that it is played with rackets and a net (the similarity ends there). A "feathered" (plastic for practice, feathers for competition... if you're good enough) shuttlecock, or "birdie", is used instead of a ball, which leads to a unique and fun variation on most paddle or racquet games.
There are two kinds of games, singles and doubles. You can also rally, though. Singles players playing doubles sometimes forget that they have a partner on the court. Heads roll. Watch out.
The ARC has some courts, and also will lend out shuttlecocks and rackets. They have a large listing of teams to play or presumably to join. There is also the UC Davis Badminton Club and the Open Gym.