The changing shapes of the moon, known as phases, are actually just changes in perspective from Earth, which appear as a change in the moon's form. From Earth, only one side of the moon ever appears illuminated by the sun, resulting its apparent change in size.
The sun provides a directional light source that strikes the moon on its visible side. As the moon orbits Earth, different amounts of the illuminated face of the moon are visible from Earth's surface. This is what is perceived on Earth as a change in the moon's shape. It should also be noted that at times the Earth passes between the moon and the sun, blocking some of the light source and casting a shadow over the face of the moon. This is known as a lunar eclipse.
There does exist an almost imperceptible change in the physical shape of the moon as a result of the Earth's gravitational pull, with the reverse being true as well. This cannot be visibly detected, but it is the reality of the gravitational forces being exerted by both bodies. The same gravitational pull also causes tidal changes in the Earth's oceans and minor fluctuations in the motion of tectonic plates.