The clicking is typically a failure of the compressor to engage. The click comes from the relay. I replaced this on my parents' fridge for like $50. If you unplug the fridge and carefully remove the relay and you shake it and it sounds like sand is inside, it means the internals are fried and it should be replaced. There are also manuals online that show how to test the relay using a multimeter.
Three things typically go bad on a fridge: the compressor (most expensive, because of the labor to evacuate and refill the refrigerant), the thermostat, and the relay. On modern fridges, there might be more that can go wrong with the electrical controls. But seriously, that clicking is probably just the relay.