The air pump is powered by a belt off the engine, like a power steering pump or an A/C compressor. It's main goal in life is to pump air straight into the cat in cars that don't have a high enough oxygen content in their exhaust to burn up the bad stuff in the cat. The additional oxygen lets the catalytic converter burn off all the extra NOx, CO, and HCx emissions.

My RX-7 came with one, a lot of domestics have them, the S2000 even has one.