A standard rappel is one in which the climber lowers herself down the mountain with her back toward the ground and her feet in contact with the rock (or snow), and walks down while letting the rope slide through the belay device.
A free rappel happens on overhanging terrain and is also commonly done by cavers. In a free rappel, the climber slides down the rope into empty space because the rope hangs free from the cliff.
In an Australian rappel, the climber faces toward the ground instead of having her back facing the ground. This type of rappelling is much more dangerous.
Soldiers often rappel differently, pushing off the surface of the building or structure they are rappelling down and "jumping" down. The goal is speed instead of getting down.
In a rescue rappel, a rescuer rappels with another (injured climber) hanging off her harness. There are several ways to rig a rescue rappel, but it places a lot more stress on the rope and climbing anchor, and should only be done for rescue.
A simul-rappel is used by some climbers to descend from rock towers where they can't set up a rappel anchor. In this rappel, the rope drapes over either side of the tower and each climber rappels off opposite ends of the tower at the same time.