There are several different kinds of air gaps in the plumbing industry. Simply put, an air gap is a physical gap (air space) between potable (clean) and non-potable (dirty and contaminated) water.
For example, Texas plumbing code states that the space between the outlet of a bathroom faucet spout and the top of the sink (basin or vessel) shall not be less than 1 inch. That one-inch gap keeps your potable water supply separated from potentially dirty water in a full sink.
However, in most residential cases the air gap we are referring to is that funny, round three-inch tall thing that makes odd noises and sometimes spews water all over your kitchen counter. Its function is to allow the dishwasher drain hose to breathe.