Quick answer: What we see as the lunar phase is the portion of the Moon which appears illuminated by sunlight. The angle of illumination changes as the Moon orbits around the Earth, thereby changing the phase.
The Moon is about 240 thousand miles away from the Earth. As the Moon orbits around the Earth, we see a complete cycle of phases taking place every 29½ days. It is easy to understand why the word “month” and the word “moon” have a common root. The lunar cycle has been recognized since the time of ancient civilizations and became an important element in the construction of calendars.
When we look at the Moon, we can observe that the portion which appears illuminated changes throughout the course of a month. As the Moon orbits the Earth, the angle of the Sun’s illumination changes and more or less of the Moon appears to be lit up. The progression has a familiar pattern which repeats in nearly the same way each cycle.
At the very beginning of each lunar cycle, the Moon is new which means it appears relatively near the Sun in the sky. During the new phase the Moon is not visible since the part of the Moon facing the Earth is in shade. About two weeks after the new phase, the Moon appears in full phase in the sky. During the full phase the Moon appears located opposite the Sun in the sky. This is why a full moon can often be seen rising in the east just as the Sun sets in the west. A full Moon appears as a fully round disk since the portion facing us is completely illuminated.
The other two important phases are called “quarters” which fall exactly midway between the full and new phases. During a first quarter moon the disk of the Moon appears half illuminated. The word “quarter” reminds us that the Moon is one quarter of the way through its cycle from either the new or full phase. Observers may sometimes get confused by the fact that during quarter phases the moon appears half lit. To avoid confusion it helps to remember that the full moon phase is the halfway point in the lunar cycle.
By observing the Moon on different days we can see the combined effect of the lunar orbit and the changing illumination. The orbital motion of the Moon causes changes in where we see it in the sky and at what times. That same motion is what changes the illumination angle of the sunlight from our viewpoint. So the lunar phase and position are intimately linked.
A common misconception is that the shadow of the Earth cast on the lunar surface causes the phases of the Moon. Of course this is not true at all. On rare occasions the shadow of the Earth does fall on the Moon. This is a special and beautiful occurrence known as a lunar eclipse. Lunar eclipses are a completely different phenomenon from the regular observation of the lunar phase.