air is used to cool the incoming air to the engine. kinda like a radiator but for air. air flowing through the cooler is used to cool the air going into the engine.
The older engines used an aftercooler to cool the air from the turbo. It wss mounted on the engine and had like a big heater core in it. coolant in the core helped cool the air at least some. The newer air to air puts air from the turbo thru a core in front of the radiator which is air cooling air, thus air to air. It cools the air much better allowing for more power etc. air to air coolers are also called charge air coolers.