Contact/Relay -- Relays are electronically controlled switches that activate the various components of your HVAC system. They manage everything from the power going to the motor to automatic dampers, humidifiers, etc. Most of them are controlled by the thermostat. The most common failure for a relay is being stuck in the "open" position. Separated from its assigned contact point, it fails to complete the connection and send the message to whatever it was supposed to operate. This failure usually occurs from use over time. Each time a relay connects and sends its signal, the electrical arcing from point to point eventually causes wear and tear.
The liquid refrigerant is returned to another heat exchanger where it is allowed to evaporate, hence the heat exchanger is often called an evaporating coil or evaporator. As the liquid refrigerant evaporates it absorbs energy (heat) from the inside air, returns to the compressor, and repeats the cycle. In the process, heat is absorbed from indoors and transferred outdoors, resulting in cooling of the building.
In addition to the information below, see these two articles for the general care and maintenance of your air conditioner: Preparing Your Air Conditioner for Summer and How to Replace Furnace & AC Filters. Most noteworthy, you should replace the filters at least twice a year, before the heating and cooling seasons. For information on furnace problems, please see Furnace Not Working.