We only see the Moon because sunlight reflects back to us from its surface. See the diagram of the full moon below.
Depending on earth's, moon's and sun's positions we see a different area of the moon being illuminated by the sun. The changing positions appear to us as different moon phases.
When the earth, the moon and the sun are almost aligned, most of the surface of the moon (as seen from the earth) is illuminated by the sun. Therefore, the moon appears as "full". This occurs approximatelly every 29.53 days