If an air conditioning unit doesn't work at all, check the thermostat, and make sure that it is set to Cool and that the temperature set on it is much lower than the temperature in the room. Inspect the circuit panels, and make sure that there are no tripped breakers or blown fuses. Reset the breakers, and replace the fuses if needed. Check if the furnace power switch, outdoor condenser’s power switch and 240-volt disconnect are all turned on.
If you didn't find any problems, put new batteries into the thermostat. Make sure to shut off power to the air conditioning unit before doing that. Lastly, open the thermostat, remove the wire in the Y terminal, and turn the unit on again. Place the bare end of the wire onto the R terminal, and hold it in place for a few minutes. If the compressor starts working in a few minutes, then the thermostat is broken and you need to install a new one. If the compressor doesn't start, call a specialist to fix it.
If an air conditioner doesn't blow air, switch off the power and gas to the unit, open the door of the air-handler cabinet, find the number on the belt, and buy the same replacement belt. Set the belt onto the motor pulley, and oil the bearings, if that is allowed for your type of air conditioning unit.