Refrigerant Metering Device -- Every HVAC system has one of these, and heat pumps have two. They regulate the amount of refrigerant going through your air-conditioner. If it fails, either too much or too little refrigerant will course along the lines and either not perform well or will overwhelm the compressor and cause it to fail. A total failure of this component will result in a complete loss of operation.
Wiring up the thermostat is usually pretty straightforward if an old one is being replaced. If a thermostat is being put in where one had never been before, that could involve running new wiring. The placement of the thermostat is also important so that its sensors can get an accurate reading of the temperature. A thermostat that is blocked by a bookcase or other large furniture will not get a good reading and will not perform efficiently.
Having the most energy-efficient system ever built won't matter much if it's not maintained. Lack of maintenance is the number one killer of HVAC systems. Before each cooling season, it's recommended that your system get a professional tune-up. However, there are things you can do in the meantime to make sure your system runs efficiently throughout the year.
Furnace Circuit Board -- The furnace circuit board controls a variety of functions. It not only handles the regular operation of the furnace, it also monitors the furnace's various safety circuits. Over time, vibrations from the furnace can weaken solder point and cause them to separate. Dirt and debris can cause short-circuits that damage the board. Failure of the circuit board can cause a variety of effects ranging from the fan not turning on to the complete shutdown of the furnace.
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.