1. Turn off the breaker to your unit.
2. Open up the small corner panel on the outside condensing unit. You will see a small, cylindrical metal thing, 2-3 inches in diameter. This is your capacity. You need to discharge any latent energy prior to messing with it. You can do that by touching a screw driver to both leads on top.
3. Take a picture of the assembly so you can know how to reconnect the new one. It's typically connected by 3 wires of different color.
4. Disconnect the capacitor and see what the size is. It will say two numbers, e.g. 70/7.5 uf. Essentially, capacitors store energy to start the two motors at your condenser; the fan on top and the compressor inside. Once the capacitor goes down, it is then unable to deliver the energy required to start the motor. That's why the fan kicks on when you start it by hand.
5. Go on Amazon and buy a new TWO. That way you have a spare when it goes out again. You could probably get them at a specialty HVAC store also, but those are more rare and sometimes won't sell to non-contractors in my experience.
6. You could reinstall the capacitor and just start the fan by hand until the new one arrives. You have to be vigilant though. Without that fan running, your compressor overheats and that is no bueno. To be safe, you could turn your AC off (maybe you have two, one up and one down and could survive with one??
7. Just make sure when you are messing with those, the power is shut off at your breaker box.

Best of luck!