The quartz in a watch is carefully carved into specific dimensions. When electricity is applied, it vibrates at a known frequency, determined in part by its specific dimensions. This is known as piezoelectric effect. That is, it is the physical property of some substances that if it is physically manipulated in some specific manner, it will produce an electric spark -- such as used in flint-free lighters; but application of an electric current will result in oscillation of the material.
The purpose of the quartz is to set the clock frequency for the watch. I is (basically) the method for how the watch keeps time, in stead of using a tensioned spring releasing its energy through a catchment device.
Basically, the pulse from the piezo element is -- via some other circuitry -- applied to a motor which will turn in increments at the beginning of the mechanical chain.