A body and its satellite in truth form a system and both rotate around their common centre of gravity. Hence both the Earth and Moon rotate around this centre - however because the Earth is significantly more massive than the Moon this can be approximated as the Moon orbiting the Earth.
The Moon does have a measurable impact on the Earth - tides are the obvious example, and tidal friction is also slowing down the Earth's rotation by a very small amount (around 2.3ms per century).
In the Moon's case, though, the Earth's gravitational pull has already effectively locked its rotation period with its orbital period around the Earth - so we essentially only ever see the same face of the Moon on Earth.
The Earth and Moon are much closer in mass than most other planet/satellite systems in the Solar System - think of the huge mass of Jupiter - and so it is difficult to show how these planets' satellites have a similar impact on the planet - but the same effects are there, albeit in much smaller scale.
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