capacitor

The first post gives you a very good explanation of how capacitor works. Typically a single phrase motor will have two sets of windings, a run winding and a start winding. Some means of throwing the windings out of phase is a necessary or the motor will not self start (a simplified explanation there are repulsed start motors as well as different resistant winding) current lags by 90° in a pure inductor, and leads by 90° in pure capacitance.

In this very simplified explanation a circuit containing induction and capacitance will be out of phase a regular motor winding (virtually purer inductance) so basically you're telling the motor which way to TURN. Capacitors can also be used increase the starting torque and then drop out of the circuit once the motor is rotating.

The subject is far too complex to discuss in a post. There are many good books available on basic motor operation. If you are trying to figure out why your garage door opener doesn't work and the oval capacitor is almost round it's worth going down in buying one to replace it. A lot of good motors get thrown away because of bad capacitors.