AC air handler buzzing sounds. Do you hear a buzzing sound when you turn on the thermostat? The only thing that comes on in the house when you turn on the thermostat is the fan relay and fan in the air handler or furnace. Try switching the thermostat HEAT/COOL switch to OFF. Then switch the fan switch from AUTO to ON. The fan (only) should come on. If the air handler makes a buzzing sound, it probably has a bad fan relay or, more likely, blower fan.
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.