Well in an induction motor, static capacitor banks are connected in parallel to the stator terminals in star or delta. It is connected to improve power factor of the motor. read more
Capacitors take active power from supply and provide necessary reactive power to the induction motor and hence motor stops taking reactive power directly from supply and power factor is improved. Capacitors in single phase induction motors are used to split single phase supply and make the motor self start in a certain direction. read more
add large electrolytic capacitors directly across the battery (or across the battery input to the PWM motor driver, or across the battery input to the digital electronics, or often capacitors in all three locations) -- these capacitors work better at supplying high currents for a few milliseconds than the battery does. read more
Capacitors across brush contacts in DC motors work as arc quenchers and save the brushes from deterioration due to continued sparking / arcing. They also in the process eliminate RF interference in nearby equipment. There is no question of connecting capacitors in series with DC motor, since capacitor acts as an open circuit for DC. read more