If you unscrew the bottom and look inside, you will probably see one or two "bladders" - a rubber round thing that is vibrated in and out by the vibrations of the motor. This is the air pump part of the pump. It works just like the plunger on a toilet plunger works - the "plunger" is pulled up and down (or in and out) and the air is pushed out and down the airline hose to your air stone.
The most common failure in these pumps is that the black rubber plunger splits and the air leaks out of the side instead of being forced down the air hose.
You can usually get a kit to fix your air pump very cheaply at the various pet stores. You need to know the diameter of the black rubber part - and get one that is the same. You unscrew the metal arm that holds it, and take off the old, split one and put a new one on.
The hardest part is getting the metal arm back and adjusted so that it pushes and pulls the rubber plunger efficiently, but does not let it vibrate on any interior surface of the pump - so it is quiet again. This takes some tinkering and diddleing, but it is possible.