I haven't had too much time to think about this question yet, but I thought I should make a note of it here to remind myself to do more research. The little I've read on the web indicates that there are two kinds of watches, those that run on a quartz oscillator and integrated circuit (powered by a battery in general) and those that are purely mechanical. Here are some good links about how quartz watches and mechanical watches work. The Swiss watch industry also has a page that shows the internal parts of quartz and mechanical watches. Supposedly over 90% of the world's watches are based on quartz oscillators.
Quartz watches are known to be so accurate that they only change by a minute over a year. But as a physicist, I know that even better time keepers exist -- that's right, atomic clocks. But it's late, so I'll report on those devices another time.
An electrical switch just works by mechanical contact. You move a piece of metal which makes electrical contact with another piece of metal, completing the circuit as you say. In a circuit model, a light bulb can be modeled as a resistor because it works by Joule heating. The heating in the resistive tungsten filament causes the tungsten atoms to emit light in the visible range.
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