The car's air conditioning system basically conditions the air, which means whilst cooling the air in the car down, it also helps to ensure there isn't too much moisture in the car, so it stops the air getting too humid. The car's air conditioning unit also cleans the air in the car by filtering the dust and other particles from the air inside the car.
The aircon relay carries electric current between the car's battery and various components that make up the car's air conditioning system and it essentially acts as an ON/OFF voltage control switch. The aircon relay is efficient and clever in that it works on very little power but produces a great many times more power than it uses.
The aircon relay will be housed in the relay box, usually found underneath the dashboard. The aircon relay connects to the car's battery, as well as connecting to the specific part that the relay is functioning, for example, the compressor clutch relay will connect to the compressor clutch.
If the car's air conditioning system has stopped working it is advisable to seek professional advice. Get the relays connected to the ac system checked, such as the aircon compressor clutch relay, to establish if it is merely a aircon relay that is broken and needs replacing or a larger problem with the whole air conditioning system.