A warm air intake or WAI is a System to decrease the amount of the Air going into a car for the purpose of increasing the of the Internal-combustion engine. This term may also be used to describe a Short ram air intake, a totally different intake modification.
All warm air intakes operate on the principal of decreasing the amount of Oxygen available for Combustion with Fuel. Warm air from inside the engine bay is used opposed to air taken from the generally more restrictive stock intake. Warmer air is less dense, and thus contains less oxygen to burn fuel in. The car's ECU compensates by opening the throttle wider to admit more air. This, in turn, decreases the resistance the engine must overcome to suck air in. The net effect is for the engine to intake the same amount of oxygen (and thus burn the same amount of fuel, producing the same power) but with less friction losses, allowing for a gain in fuel economy, at the expense of top-end power.
Opposite principle of a Cold air intake (CAI) which significantly differs by collecting air from a colder source outside of the engine bay.