Hi cai! Yes it does! All massive bodies have gravitational pull on each other. You are in fact pulling on the Earth just as it is pulling on you! Gravity works such that two bodies will orbit each other around their center of mass. In the case of you and the Earth, the Earth is sooooo much more massive that the center of mass is almost exactly the center of the Earth. So you don’t notice your mass at all. The moon is pretty massive, but not nearly as massive as the Earth. So, the center of mass of the Earth-Moon system is very near the center of the Earth but not exactly. So the Moon and the Earth both orbit this imaginary point. However since the center of mass is in the Earth and near its center the Earth doesn’t move all that much, so we only really notice the moon’s movement.
A different extreme is the case of Pluto and its moon Charon. They are very close in mass: Charon is half the size of Pluto! So instead of Charon orbiting Pluto, both Charon and Pluto orbit each other – the center of mass of the Pluto-Charon system is offset!