As you can see from the label above it is an Emerson mini fridge with product code CR519B (a link to the product manual http://www.emersonradio.com/resources/owner_manuals/manual_1313100237.pdf).
Whenever I had tried plugging in the refrigerator there was an audible clicking noise as the compressor attempted to start. The clicking would repeat itself about three times before it stopped. A Google search on the symptoms suggested that the compressor start relay was burnt out.
The label for the compressor, which seems to be a fairly generic compressor and fortunately was not the cause of the problem.
The compressor start relay and overload protector are attached to side of the compressor and covered by a plastic cover held in place with a metal clip. Removing the clip was fairly simple and just required a bit of leverage with a flat head screwdriver.
Removing the plastic cover revealed the overload protector and the PTC (positive temperature coefficient) start relay. You can see them attached to the compressor pins in the photograph on the left. They are separated from the compressor in the photo on the right, where the overload protector is the item attached to the beige clip on the bottom and the PTC relay is attached to the blue clip to the right.
As I had removed the relay components from the compressor I took the opportunity to measure the resistance of the compressor windings. Apparently the way these sorts of compressors work is that there are a two set of motor windings for starting and running the compressor (as starting the compressor from a standstill requires significantly more power that it takes to keep the motor running continuously). The resistance between each of the compressor pins is noted in the diagram on the right. As the resistance between the two bottom pins is roughly equal to the sum of the resistance between both pins and the top pin this shows that the compressor windings are neither short-circuited nor open. This is fortunate as a faulty compressor tends to doom refrigerators to the junkyard (compressor repair is generally more expensive than buying a new fridge).
I then tested the overload protector and found continuity between the pin where it attaches to the beige clip and the socket for the top compressor pin (the pin socket is the metal bit in the center of the photo on the right). This indicated that the protector was working properly (which was logical as the protector closes the compressor to try to start itself but opens the circuit when it heats up - which happens in the case of repeated failed starts).
That left the PTC relay as the likely source of the problem. The PTC relay contains a thermistor that initially allows electrical current to flow across the start windings of the compressor. As the thermistor heats up its resistance rises along with the temperature (hence positive temperature coefficient) until the current no longer flows through the start windings, flowing instead through the run windings (which is important as having current flow through the start windings for too long could possibly damage the compressor).
I used a multimeter to test the resistance between the two pin sockets of the PTC relay and read an astoundingly high value of more than 2000 ohms, proving that this was the faulty component. Upon shaking the relay I heard a faint rattle (this is another one of the telltale symptoms of PTC relay failure as I had learned in my research on Google). Curious, I pried open the relay with a small flat head screwdriver and discovered the reason why.
The PTC thermistor inside the relay was cracked (with the loose pieces making the rattling noise) and seemed to be worn where the disc was held by the metal contacts.
Fortunately the PTC relay is very inexpensive to replace and I was able to order one from Amazon for $2.41 (including shipping) from China. Had I been in a rush I could have ordered one sooner from a closer location for a significantly higher sum, but I had the liberty of time. The replacement arrived in a few weeks and was near identical to the original (the two photos above of the new relay should demonstrate the similarity to the old one). It was then just a simple matter of putting everything back together and testing the refrigerator to make sure that it worked. All in all I think this is one of my favorite DIY repairs for its simplicity and the sheer cost-effectiveness of replacing the relay versus buying a new mini refrigerator (not to mention the simple environmentally-friendly pleasure of saving this appliance from the junkyard).