What does an air pump do?

It pumps air. On most cars it either is the source for air for the air ride suspension, or it is an emmissions device that adds air to the exhaust.

The "Air Pump" actually AIR means "Air Induction Reaction". Its job starting in the late 60's was to inject air into the exhaust system in order to burn off excess unburnt fuel thereby reducing Hydrocarbons.