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.
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.
Because an air conditioner moves heat between the indoor coil and the outdoor coil, both must be kept clean. This means that, in addition to replacing the air filter at the evaporator coil, it is also necessary to regularly clean the condenser coil. Failure to keep the condenser clean will eventually result in harm to the compressor, because the condenser coil is responsible for discharging both the indoor heat (as picked up by the evaporator) and the heat generated by the electric motor driving the compressor.
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.