Compressor starting circuits for household refrigerators vary depending on age and manufacturer. Almost all employ a relay and overload. Some use a starting capacitor as well. Regardless of the configuration, they all serve the same purpose…to start the compressor and get it up to its rated rpm and provide over-current protection. It may not be important to know exactly how the circuit operates as long as one knows that if the starting circuit is defective the compressor cannot start.
Compressors are hermetically sealed electric-motors that pump refrigerant vapor. Like any electric motor they draw more current when starting than when running at their rated speed. Thus compressor motors have two windings, a start winding and a run winding. At approximately 75% of the rated speed the start winding is taken out of the circuit (by the starting circuit relay) and the run winding keeps the compressor running. The windings are connected to the starting circuit by a three-pin connector on the side of the compressor.
The most common configuration is illustrated below. Again, depending on the age and manufacturer this arrangement might not apply.
The first refrigerator repair I ever did was to a compressor starting circuit. Unfortunately by the time I discovered the defective seven dollar part my mother had already bought a new refrigerator. It did make me realize that a very inexpensive part can cause a refrigerator not to work. Over half the Haier refrigerators donated to Habitat for Humanity by a national appliance retailer had defective compressor starting circuits from the factory. Most of the rest were damaged in shipment or had defective evaporator fan motors.
Keep in mind that the power to the compressor starting circuit comes through the cold control and defrost timer (if the unit has automatic defrosting) so if these parts are defective there may not be power available to the compressor. That said…most evaporator fans are wired to run when the cold control makes and most condenser fans are wired in parallel with the compressor. Therefore if the fan in the freezer compartment is running and the condenser fan on a forced-air condenser is running the compressor should be running.