In a quartz movement watch, a battery sends an electrical signal through a quartz crystal. The crystal vibrates when electrified, producing oscillations as it vibrates that drive the motor in the watch.
In a mechanical watch, a person has to manually wind the watch. The act of winding coils up a spring, called the mainspring, which coils up and stores potential energy. As the spring unwinds, it releases energy through a series of gears and mechanisms in the watch, providing and regulating power over time.
In an automatic watch (also called a self-winding watch), the watch has free-spinning rotors inside. When you wear the watch and move your arm around, those rotors spin in response to the movement, and transfer their rotation to the mainspring. The spring unwinds as in a mechanical watch, but instead of being deliberately wound by a person, the watch is simply powered through your own natural kinetic energy.