Compressor won't run or it is "clicking' on and off:

This normally could be a bad start relay and over load for the compressor, dirty condenser coils need cleaning, condenser fan motor is not running or the compressor itself. To check the compressor properly you should test it with an amp meter, each compressor is rated in running amps  ( see model/serial tag or the sticker on the compressor )...if the compressor is drawing too much current, this may be why it is shutting off. Check the model tag for proper amp rating. Some relays will be easy to notice the broken with them, they may be burnt.  Some of these relays can be removed and give them a little shake, and if they rattle around inside the relay is likely bad. Compressors can also seize and click on and off or one of the electrical windings inside the compressor could have opened up. You can use a test cord to help check the compressor and this also will help with the amp test. If the compressor checks ok and you want to install a replacement start relay. You will need to find out which winding is which. You will need an ohm meter to determine the windings. First check to see if you have windings in the compressor. Then check to see if they are grounded. If the compressor has windings and they are not grounded, you can find which winding is which. Check for grounded windings by reading from each terminal to a good ground on the cabinet. Read from the top terminal to the lower left terminal. Read from the top terminal to the lower right terminal. Read from the lower left terminal to the lower right terminal. Write down the resistance of each reading as you go. The highest reading you get will be the run and start winding of the compressor in series with each other. The other terminal left will be the common terminal. Read from the common terminal to each of the other terminals. The terminal with the lower resistance will be the run winding. The higher resistance the start winding. Using the original style relay is always preferred, but using a general replacemen can "get you by" if needed.