The ABS relay manages the power supply to the ABS system. Relays generally handle the switching of circuits in different systems in the car.
Relays allow high power circuits to be switched safely. Certain parts of a car require lots of current to power them (eg the ABS system) the relay is capable of switching this via a lower power circuit.
Relays will degrade over time and can cause problems as the systems they control appear to fail. They're usually very easy to replace.