What is the purpose of the Motor Run Capacitor?
It's more than likely a single phase induction motor. 

Unlike a 3 phase motor that can automatically generate a rotating magnetic field, a single phase induction motor has to manufacture one by using 4 stator poles. Two of them are fed with the regular AC voltage and the other two are fed AC via a capacitor. If the value of the capacitor and winding inductance is chosen to electrically be a low Q resonant circuit at the AC frequency supplied, the voltage phase difference between the two sets of stator coils will be approximately 90 degrees and this will set the motor spinning. Here is a decent link and below is a snapshot: -