The ABS (anti-lock braking system) pump relay manages the power supply to the ABS pump. Relays generally handle the switching of circuits in different systems in the car.
Relays allow high power circuits to be switched safely. The ABS pump needs a lot of power to operate so the pump relay manages the supply to it.
Relays will degrade over time and can cause problems as the systems they control appear to fail. They're usually very easy to replace.