Capacitors are the only way to make a rotational magnetic field when you have single phase power. Now what that means in laymen terms is basicly anything that relies on spinning needs a capacitor to properly and consistently run. Meaning any motor like fan motors and compressors will require a capacitor if they are powered with single phase.
When you give power to the wires in a motor on single phase equipment they are split on a cross section of the shaft by equal power coming in from two side 180 degrees each. With that equal power it is hard for the motor to know which direction to start the spinning. When you hook up a capacitor to the line it absorbs some of the electricity on one side of the shaft and releases it very quickly. This allows one side to be stronger than the other thus beginning the rotational magnetic field.