What does the earth rotate around?
The Earth-Moon Barycenter.
The Moon has about 0.0123 of the mass of the Earth. This means that the Moon doesn't orbit around the Earth. In fact the Earth and the Moon orbit around their centre of mass which is called the Earth-Moon barycentre.
The barycentre is about 4,700 km from the centre of the Earth which is still inside the Earth as the Earth's radius is about 6,400 km.
A consequence of this is that the Earth doesn't orbit around the Sun. It is the barycentre which orbits around the Sun.
To take it once stage further, the Sun and all of the planets actually orbit around the solar system barycentre which is the centre of mass of the whole solar system. This is a point just above the surface of the Sun.