How does a wristwatch work?

You have a mainspring wound up tight inside a small barrel. The outside of this barrel is toothed like a gear and has other gears riding up against it. Call this the "gear train". The sizes of the other gears in this train make one hand go around the dial once every 12 hours, one every hour and one each minute. Left with just this the hands would whiz around too fast to see as the spring unwound in a big hurry. to prevent this you have a little wheel, a "balance wheel", with a tiny "hair spring" wound around it. This wheel swings once to the right and coils the spring up, then one to the left and the hairspring uncoils. It continues back and forth. There is a tiny lever connected to the middle of this wheel which gets pushed back and forth. Each time it moves the gear train is allowed to move just a little bit. This keeps the gear train from running away under pressure from the mainspring. The speed of the balance wheel's back and forth motion is what determines how fast or slow the watch is.