Dear Student,
There are usually two capacitors in a ceiling fan motor. One is the "start" capacitor and the other is the "run" capacitor. Simply put, an AC motor needs a rotating magnetic field in order turn the motor shaft (fan blades). This is done by applying voltage with different phases to different windings. In a single phase system there is only one voltage phase. The capacitor is used to provide a phase shift (i.e. a time offset between currents) in the windings of the motor, making it appear that the motor is operating in a multiphase system.
Regards