The components on the side of the compressor are called your relay, starter and capacitor

The clicking noise is very distinct and it will sound like it's coming from under the refrigerator or from the back. You may also hear buzzing. The overload
Now, there are a few other things that will cause this to overheat and shut off. One is if your condenser coils are covered in pet hair and haven't been cleaned in some time. Lint, dust, hair, who knows what else can be there but you get the point. If those coils are not kept clean, the compressor will run hot. That will cause the overload on the side of the compressor to overheat and "click"-no cool. Unplug the fridge and clean the coils. This may fix the problem if the overload or compressor weren't damaged. You will know right away. If the fridge runs fine after cleaning the coils then you are okay. You will know usually within a few minutes if the something has been damaged. It will still click.
Another thing that can cause the click is an adaptive defrost control board. It contains a relay that sends power to the parts on the side of the compressor. When the defrost board doesn't send full power to the compressor, this causes an increased amp draw to make up for the voltage drop. Increased amp draw equals "CLICK"
One more thing that may cause this clicking sound is the condenser fan not working. The condenser fan is the fan that is next to the compressor. It has a couple jobs. Evaporate the defrost water and keeping the heat off the condenser coils and compressor. If the fan isn't running when the compressor is running, guess what will happen. "CLICK" I think you know why by now, but just in case, the compressor overheats and the overload on the side of the compressor, protects it by shutting off the power. Hopefully if your fan motor was bad, the overload did its job and saved your compressor. You can place a separate fan on the floor blowing on the compressor and coils to temporarily get you by until your new parts arrive.
Okay, you cleaned the coils, got a fan if you needed one and it still clicks, now what? The first thing to do is replace the components on the side of the compressor. Hopefully the compressor starts up and the new components don't click like the old ones. If they don't click you are good to go. Good job, you just saved a bunch of money on appliance repairs.
But what if it still clicks? Well, to put it nicely, your refrigerator may need to find a new home(scrapyard). That means your compressor has internal damage or you have some sort of freon restriction. If your refrigerator is less than five years old it may be covered by the manufacture's limited warranty. That means this repair will not cost you. But, if it's older than that or you don't have some sort of extended warranty plan, this repair will cost you at least $300. And if you are really unlucky it could cost around $600. The difference in price being whether you only need freon work, or you need the compressor replaced.
I think I've covered the basics. You now know what the annoying clicking sound is when your refrigerator gives up and stops getting cold. The clicking is capable of happening periodically. Make sure the refrigerator isn't running in temperatures above 85 - 90 degrees F. Don't let the refrigerator be exposed to direct sunlight. Keep the coils clean. Don't leave the doors hanging open for long periods of time. If you are having a defrost issue this sound could be heard. That is because with a defrost issue, the compressor runs constantly and that will cause it to overheat.
If you are having a Defrost Issue be sure to read about Defrost Problems on my blog. Search for Frost in my Freezer.
Thanks for taking the time to read this. If you have more questions about this I can be found on YouTube.com/ApplianceOutlet. Post your Comments or Questions either on this Blog or at the bottom of my YouTube videos.