The AC compressor in a car drives the air conditioner. Unlike an AC compressor in the home, it is not driven by an electric motor. Instead, it is directly driven by a belt attached to the engine. An AC compressor uses a lot of energy, so it is not practical to run the compressor all the time. The AC compressor clutch serves to turn the air conditioning on only when it is needed.
Compressor clutch basics
A belt drive attached to the engine spins constantly, providing energy for the AC compressor. The belt turns a rotor, which has a friction plate attached to it. Near the rotor is an electromagnet -- a coil of wire which produces a magnetic field when current is run through it. Across from the friction plate is another friction plate attached to the rotor that powers the air conditioner.
When the electromagnet turns on, it produces a powerful magnetic field through the rotor. Since the fan belt rotor and the air conditioner rotor are both made out of iron, the magnetic field magnetises both of them, pulling them towards each other. This causes the friction plates to push tightly in to each other.
Initially, the rotor friction plate is turning and the compressor friction plate is not. Once they are touching, however, friction between the two plates makes the air conditioner rotor spin. When the electromagnet is turned off, the friction plates spring apart and the compressor stops moving.