I agree with beenthere. You need to determine where the electricity is and isn't getting to. If it is a bad capacitor which would mean that the unit is receiving power into it and across the contactor when the stat is set for cool and calling, you will need a digital multimeter to test that cap and just so it is said, please make sure you turn the power OFF to the unit, not just by way of stat but by breaker or disconnect, before you go messing around.
A capacitor can only properly be checked with ALL wires disconnected from it so write down what wires are on what what terminals and get a good mental image of them before you remove any of the wires and with a digital multimeter that reads mfd aka microfarads you will check the cap. Once all wires are off of the capicitor, the cap needs to be discharged by sticking a screwdriver across the poles. Test across herm and com and test between fan and com and get your readings. You have to hold the leads on the terminals for a few seconds to receive your readings. The proper mfd numbers should be labeld on the cap already to match your readings against.
Nothing should ever read between herm and fan as if it does than the cap is bad, if it's a dual run cap meaning both the compressor (herm) and fan are running on one capacitor.
Now if your contactor is not pulling in when the stat's set point is calling for the system to come on in cool than you have a different problem which any of us on here will assist you with as needed. You know how to tell if a contactor is in or out or if it's good or bad? I can tell you but I might have to charge you for that class.
Good luck and ask away.