When your air conditioner compressor hums but the fan isn't moving, this is usually the result of an electrical issue preventing your system from working the way it should. The problem can typically be narrowed down to fuses, capacitors, and fan motors.
Bad Fuse or Fuses
Your air conditioner's fuses act as a protective measure against electrical damage, such as surges, that can damage expensive components. Fuses are relatively inexpensive, which means they can take the brunt of electrical trouble and be easily replaced.
If your fuses are old or damaged, your air conditioner may still be receiving power but not enough to actually start. You can test your fuses to verify whether they're the problem or not, or you can swap them out with new fuses to see if that fixes the issue. Before you test or replace your fuses, make sure to shut off your air conditioner's power at the breaker pane to prevent any chance of injury.
If your fuses are old, or if you have had electrical problems in the past, it's also a good idea to keep spares on hand. Replacements can run out of stock during the hottest months of the year, which can cause delays when you need your air conditioner most.
The start capacitor is responsible for providing enough power to jump-start your fan motor. Without a working capacitor, your fan blades won't move even if your motor itself is working fine.
One way to test if this is the case is to use an object like a stick to manually get the fan blades moving while you hear the humming sound. If they start to move on their own and run fine until the system shuts off again, the problem is most likely the start capacitor, which will need to be replaced.
Using your air conditioner with a bad capacitor can cause strain on your system, so even if you can manually get it working in the meantime, schedule a repair visit as soon as possible.
Bad Fan Motor
In rarer circumstances, the fan motor itself could be the problem. This is more likely the case if your air conditioner is more than a decade old, as this is when more important parts can start to fail.
If other troubleshooting methods don't work, your motor could still be receiving power, but it won't be able to run correctly. In the event the motor is broken, it will need to be repaired or replaced. The good news is that a fan motor can usually be replaced by itself, meaning you don't need to worry about having to replace your whole air conditioner.
Contact an AC repair contractor to learn more.