Clocks, in general, can be fashioned out of any oscillatory phenomenon as long as it is regular enough. Initially these were mechanical: pendulums and springs, and so on. For quartz watches, this is still the case, and the oscillator is a tuning-fork-shaped quartz crystal that vibrates mechanically, though this vibration is read by an electronic circuit.
Regardless of the specific implementation, any clock (be it a pendulum clock, a quartz watch, or a caesium atomic clock) is simply an oscillatory system with a bunch of 'readout' apparatus around it. This oscillatory system will have a base frequency but it will also deviate from it to some extent. (And yes, this includes atomic clocks, because of things like magnetic noise.) One simply accepts these deviations, in the understanding that they set the limit to which the clock can be trusted. For quartz watches this isn't great, as they can slip by a few seconds every month or so. For atomic clocks this is more like one second in 300 million years, but it is still finite and it still means there are things one might want to do that are not yet possible because we do not yet have accurate enough clocks.
Finally, there is the issue of calibration. Once one has an oscillatory system, one needs to adjust the frequency to match, as well as possible, all the other clocks around. To do this, watch factories will have stable reference clocks (which you can assume are atomic clocks) which they use to calibrate their resonators. If they're too fast or too slow they are trimmed down to bring them to speed, and this is done until they are as accurate as their finite precision allows them to be.