Manual watches require a movementi of a series of gears to "tick" in increments of time. These ticks are seen as movements of the hands on the face of the watch. A rotori in the watch sits on a staff in the middle of the watch's movement. It rotates in a circular motion and winds the mainspring which is the source of power in mechanical watches.
Automatic watches on the other hand while using the same gears, derives its source of power from the winding of its spiral spring. Any movement from the arm or wrist rotates the rotor which in turn winds up the watch. To maintain an automatic watch well, you will still need to manual wind it at least once every two weeks to keep the watch well lubricated and its timing precise.
Some automatic watches have a power reserve feature. This feature lets the movement of your watch keep time for anywhere between 10 and 72 hours. Hence, the bigger the reserve, the longer your automatic watch will keep running without further movement or manual winding.
Quartz watches differ widely from a manual or automatic watch winding system. Its probably invented in the 19070s using the power in batteries to send electrical energy to the quartz crystals or silicon dioxide in a watch. Quartz crystals have a unique property that conducts a constant pulse on its surface when bent or compressed. This phenomenon is called the Piezoelectric effect. 
It is this constant pulse that gives quartz watches its precision right down to hundredths of a second. To convert the pulses into a steady power that can move the watch hands at regular intervals, this impulse is simply passed through a stepping motor that converts the electrical energy of the pulse into mechanical energy to run a watch.