It is only when you get an gigantic number of atoms together, as in the case of the planet Earth, that the force of gravitational attraction is significant. The reason why gravity on Earth never changes is because the mass of the Earth never changes.
Subsequently, does Earth have the most gravity?
So, to begin answering your question, Earth has a greater gravitational pull than the moon simply because the Earth is more massive. The moon’s gravitational pull also affects Earth, though; the rising and falling of the tides are an effect of the moon’s gravity.