The A/C compressor relay is the switch that engages the air conditioning compressor. When you press the A/C button on the dash, the engine computer sends a low-current signal to the relay. This activates a coil inside, closing the contacts to send 12-volt battery power directly to the compressor clutch electromagnet. Without this relay, the high amperage drawn by the clutch coil would burn out the dashboard switches.
Relays fail due to worn contacts or burnt internal windings. If the relay sticks open, the compressor will never turn on. If it sticks closed, the compressor clutch will stay engaged constantly, draining the battery even when the vehicle is turned off.