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".
Moon phases diagram
Moon phases diagram/image credits: StarChild team at Nasa
As seen in the diagram, when the earth is between the moon and the sun, then we see a full moon.
When the moon is between the earth and the sun, we have a new moon. If they are perfectly aligned, then we have a solar eclipse.
Duration of each of the moon phases
Each of the 4 lunar phases (new moon, first quarter, full moon, last quarter) lasts approximately 7 days (~7.4 days), but varies slightly due to lunar apogee and perigee.
A complete circle, e.g. full moon to the next full moon, takes approximately 29.53 days.
For more precise timing, there are moon phases calendars ( and of course our moon phases screensaver) with precomputed dates for the full moon and the other phases.
Useful moon phases software
Installing the free edition of our screensaver to see the current moon phase whenever the screensaver starts. It will also notify you about upcoming full moons and new moons.