A word about gravity:
Gravity is not unique to Earth or to any planet or heavenly body. The idea is that mass pulls mass. Two boats that are close enough will pull each other. Earth has more gravity than the moon and Jupiter has more than Earth.
The idea behind any planetary or satelite orbit is that the gravity of the center planet pulls the outer, orbiting object toward itself, while the centrefugle force of the orbit keeps it from going in to the center. That’s how Earth and all other planets of our galaxy orbit the sun.
Now, since the moon is only 6 times smaller than Earth it has an impact on Earth. Therefore, not only does the moon orbit around Earth, but Earth orbits around the moon as well, albeit in a small and not so noticeable way.