In contrast to a mechanical watch movement made from gears and springs, a quartz movement comprises a circuit board, a quartz crystal and a battery. The battery charges the quartz crystal oscillator, which drives a motor to move the hands at a constant rate. It will also power any other functions the watch may have. You can usually tell a watch is a quartz from the dial because of the way the seconds tick rather than sweep (though there are brands who add in the tick, but that’s another piece entirely).
As with mechanical watches, there are hierarchies – of quality, complication and price. There’s standard quartz, which beats at 32,768hz; very high-precision calibres (VHP) which are more precise, and hybrid meca-quartz – a rare breed usually found in chronographs and means the watch functions are battery powered but the stopwatch functions are mechanical. With this you get all the accuracy of quartz but with the kudos of gears and cogs – a win-win for some.