Why do we see phases of the moon?
The phases of the moon, as seen from the earth, are a result of the angle the moon has in relation to the sun. A fact we must remember is that the moon does not have its own light, thus what we see is sunlight reflected from the surface of the moon. The moon revolves around earth and completes one cycle in about a month (around 30 days). During this revolution, a combination of the moon's angle with sun and the reflection of sunlight from moon's surface causes the phases.
The moon's phases start with the new moon. After that, the moon starts (as it appears from earth) to grow, or wax, as it moves away from this position, and we see progressively more of the sunlit side of moon. We move from new moon to one-quarter moon and then to full moon (when we see the complete sunlit side of the moon). After that, it starts to appear thinner, or wane, with each day and crosses over from three-quarter moon finally to new moon again.
Some people wrongly think that the new moon is an eclipse of the moon, when the shadow of earth falls on the moon because of sun-earth-moon alignment. This is not true. The reason it is not true is because the moon has an orbit that is not on earth's ecliptic, or orbital plane. There are between four and seven eclipses of the moon each year, and there are 12 new moon phases, so eclipses and new moons can't be the same.
Note that irrespective of what we see from earth, the moon is always lit by the sun, it's just that we are unable to see it fully because of the angles.
Hope this helps.