Evaporators include humidifiers, which boil small quantities of water to create steam to increase the humidity in a room, and the part of a refrigeration system (phase-change heat pump) which cools the air moving through the system.
Inside the air conditioning evaporator, low pressure liquid refrigerant is heated by the air flowing over it, which in turn is cooled. As the refrigerant is heated, it changes state (of matter) and becomes a low pressure gas. It takes a considerable amount of energy to change the state of matter, which along with the fact that liquid is more dense and thus provides more cooling than gas is the reason why air conditioning systems provide so much cooling with only limited surface area. Condensation later occurs, appropriately enough, in the condenser.
In home and business air conditioning systems, the evaporator is usually located inside a vent somewhere. In automotive systems, it is sometimes inside a plastic box that also houses the heater core, that is bolted to the firewall inside the vehicle's cab. Often it's in its own box in the engine compartment, bolted to the other side of the firewall. This makes replacement much easier.