Most small single-phase compressors have a start winding and a run winding. The starting relay, either a current-type relay or a PTC thermistor device, connects the start winding for the first moment to give the motor a strong turn, then disconnects it once the motor is up to speed. Without it, the motor hums but will not start.
The overload protector is a bimetal disc that senses both current and temperature. If the motor draws too much current, for example during a locked-rotor attempt, or the compressor shell overheats, the disc opens and cuts power. Once it cools, it resets automatically and the compressor tries again. The repeated click-hum-click sound is a classic symptom of a starting problem.
These parts are inexpensive and are often the first thing a technician tests. But a tripping overload is usually a symptom, not the cause. Low voltage, a dirty condenser, a failing start capacitor or an overcharged system can all make the overload trip. Replacing the overload without finding the cause leads to a burnt compressor.