My Comfortmaker AC compressor outside makes this strange sound when I turn it on, and the fan in the compressor doesn't run. https://vocaroo.com/i/s0rx9CsN4XaQ Every few seconds it also makes the sound you can hear at 0:06, which turns off after 1 second, and this cycle keeps going until I turn the system off. Any help you can provide on what the problem could be would be greatly appreciated!
Given that you are likely talking about a residential air conditioning compressor with a single phase motor, that sound would be indicative of a failed Starting Capacitor in the compressor motor, so the motor is not able to start. Many of those are hermetically sealed compressors, so the capacitor is separately mounted somewhere away from the motor itself, then there is a "Potential Relay" that takes it off-line once the motor has started (as opposed to a centrifugal switch found on standard single phase motors). So if the capacitor is OK, it might also be a failed potential relay. Either way it's not something the average homeowner can fix, it takes some specialized knowledge of what to look for and how to troubleshoot it. I suggest calling an HVAC service tech.