A wound-up mainspring drives an escapement, which drives an oscillator.
A mainspring is simply a spiral of metal, which is wound by twisting the center or outside and so coiling it more tightly. This is what provides energy to the watch, and typically will be hooked up through gears to the hands of the watch. As such, the unwinding of this spring is what moves the watch hands.
The escapement prevents the spring from just spinning freely and unwinding. It is combined with the oscillator – it gives the oscillator a push to keep it going, and in turn the oscillator ‘tells it’ when to move forward. The oscillator keeps time, the spring keeps things moving, and the escapement is the bridge between the two.
This makes more sense with an animation, so [here](https://upload.wikimedia.org/wikipedia/commons/a/a7/Anchor_escapement_animation_315x478px.gif) is an animation. This is the escapement for a pendulum clock, but it is conceptually the same. The mainspring drives the big spiky wheel (in a pendulum clock the spring is usually replaced with a weight), and the pendulum swings side to side as the oscillator to keep time.
In a watch, the oscillator is a weight on a spring, bouncing back and forth (typically, a weighted wheel, spinning back and forth), so that it does not rely on gravity.
Edit: minor correction for terminology
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