[fintstone]

As noted, most A/C issues where the compressor does not run is the start/run capacitor...although a bad compressor or the compressor thermal cutout switch (in the compressor will keep it from starting). In my experience, a bad compressor typically pops breakers as it shorts to ground...so more obvious.

The caps (usually one for the compressor and one for the fan) are easy to test and often you can tell just by inspection (bulges or leaking). If the fan doesn't run, the compressor may cut out (thermal cutout). It does not reset until the compressor is cool.

You likely have a 120v contactor of some sort. It is just a big relay. When on, you should have 120V on the input and the output (as measured across the input/output leads). I start there with an A/C unit where nothing works. The contactor in most home units (they are 220) allow you to manually close the contacts which should turn on the fan and compressor while you hold it.