A self-winding watch
is simply a watch that winds itself. More specifically, it is a mechanical
watch that uses the natural movement
of your arm and gravity
to power itself. No batteries
or manual winding is necessary.
Typically, the watch will have a small weight attached to a gear. On my watch, the weight is a semi-circular disc attached to a central pivot. The weight will pivot with each small movement, winding a spring.
Of course, if you leave your watch off for long enough, it won't be able to wind itself, and it will stop. Most watches can store up enough energy for a day or two.