What is the force that pulls the moon toward earth?

There really isn't a force that uniquely pulls the moon toward the earth. Gravity is the force, and the force is the mutual attraction between the earth and the moon. In a very real sense, the earth and the moon are orbiting each other, and it is not just the moon orbiting the earth. However, from our vantage point on earth it is easy and convenient to think of the moon as the object doing the orbiting.