Another important aspect of air conditioning is that it dries the air, speeding up evaportation of sweat, which makes you cooler.
In addition to what @stanleybmanly said, there are two separate sides of a A/C system: The high pressure (hot) side and the low pressure (cold) side. The high side is ½ of the compressor through the condenser, then the receiver/drier, up to the TEV (thermostatic expansion valve). The pressure goes from high to low at this valve causing the refrigerant to get very cold and begin taking on BTUs. Then the refrigerant goes through the evaporator which cools and dries the air. Then the refrigerant goes back to the compressor and the process begins again.
Have you ever witnessed water dripping under your car in the summer? That is your sweat that the A/C removed from you and the inside of your car.