There are multiple methods to measure time. In the past, we had unreliable time measuring tools like sundials and hourglasses. Then Christiaan Huygens using the concept of SHM invented the pendulum clock which was the first precise enough time measuring device. Eventually Pendulum was replaced by Quartz Crystals which are used in Analogue clocks and watches. I want to know which method do digital clocks and watches use? It is Quartz crystals or something other? Is that other more precise than Quartz crystal?
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The quartz crystal is pretty much the standard for any electronic clock or watch.
So the timekeeping circuit in a modern digital watch will be essentially the same as that in a 1980’s analog quartz watch, only taking that time signal and displaying it on a digital display rather than triggering the stepper motor running the hands.
And even when you look to the more modern synced devices, the timekeeping will still be based on a quartz crystal keeping yonder just the same – the difference will be the fact that periodically your watch will sync with the radio time signal, a GPS satellite, or using Bluetooth to sync with the internet. These signals are just used to adjust the normal quartz clock to ensure it stays accurate and is only able to drift a small fraction of a second over the course of a day.
There are a few other systems, but nowadays are in limited use. For example a lot of alarm clocks and appliances used to keep time based on counting the frequency of the voltage of the power supply they were plugged into, and there were systems based on electronic tuning forks and similar that have mostly fallen out of favour (Bulova being one brand I believe still produces them).
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