Watches show the current time, and are useful for gauging how much time is left before sunrise or sunset. They can commonly be found in residential areas. Press "O" to bring up the watch, or double tap "O" to have it stay on your screen.
When playing with others, it's possible for your game time to not be correct. A watch can help identify if your time is out of sync: if the time on your watch doesn't match what others have, someone is desynchronized. It can be fixed by just reconnecting to the server. Usually this doesn't matter, but sometimes it can still be daytime for some players, but night for others.
It's possible to find north using a watch. This video explains how to do so.