An Aquarium air pump is used in your aquarium for several reasons.
The first and most important is to cause bubbles to irritate and break up the the surface of the water.
This is called gas exchange. At the water surface carbon dioxide is released and air is taken in to the water.
The movement of the water causes air to be trapped in the water.
This air can then be extracted by the fish from the water using their gills.
The second use of your air pump is to draw water through your box filter, under gravel filter, protein skimmer or sponge filter by pushing air through the filter at one end and releasing it at the other, this movement of air causes water to be pushed through the filter in the same fashion and is the primary way that your water is cleaned and kept particle free.
In some tank setups air pumps can also be used to drive your aquarium ornaments or air stones.
Always choose an fish tank air pump that is a little bit more powerful that you think you will need since you may want to run an extra airstone, or aquarium ornament at a later stage.