Start capacitors are used in the electric motors start winding's to provide enough current to get the motor started.
The start capacitor gives the motor a push to get the motor running and without a motor having a capacitor to boost the power during the startup the motor would just hum and not get started.
A run capacitor is used to boost the current that is used to provide power to the motor and as the name implies keep the current constant to the electric motor to keep the electric motor turning at a constant speed.
If a start capacitor on a motor goes bad it will not allow the electric motor to start and the motor will just hum.