The stars are stationary, and it is the earth that moves in its rotation, making it appear as if the stars are moving.
If you are referring to shooting stars, these are actually not stars but meteorites that are traveling close enough to the earth for us to see them. They are called “shooting stars” because they look like stars, but they aren’t. Stars, as the previous poster asserted, are stationary (like our sun).
No stars are stationary, and neither is the Earth. The stars orbit around the centers of their galaxies, which in their turn move relative to each other as the universe is expanding. Stars are far enough though, so their movement does not contribute much to the observed motion across the sky – that one’s mainly due to the Earth’s rotation around the Sun.
The amount that we perceive the stars actual movement (e.g. in our galaxy’s rotation) is very small and almost impossible to notice with the naked eye. You’ll notice two types of motion in the stars across the sky: across or around (daily) and up and down (seasonal). The across motion is due to the earth’s rotation on its axis, which points up at the North Star in the northern hemisphere. The stars we see seem to revolve around the North Star, but their motion is called “relative motion” because even though your eyes are spinning about the earth’s axis, you feel stationary. In a long-exposure photo, the motion looks like this:
Where the center of the circle is the North Star.
The other type of rotation, up and down, is a result of the earth’s revolving around the Sun. The earth’s axis is tilted, so at some points in its revolution it is tilted toward the Sun, and other times it is tilted away. This creates a slow wobbling of what is called the Ecliptic Plane, which is the path the Sun seems to take across the sky, and the wobbling is why in summer it seems like the Sun rises higher in the sky than in winter (in the northern hemisphere). The same effect happens with the stars: different constellations rise above the equator in winter, and others in summer.
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