When one observes the phases of the moon he is simply observing the moon's day and night, a natural shadow from the sun illuminating half of the spherical moon at any one time.
The lunar phases vary cyclically according to the changing geometry of the Moon and Sun, which are constantly wobbling up and down and exchange altitudes as they rotate around the North Pole.
When the moon and sun are at the same altitude one half of the lunar surface is illuminated and pointing towards the sun, This is called the First Quarter Moon. When the observer looks up he will see a shadow cutting the moon in half. The boundary between the illuminated and unilluminated hemispheres is called the terminator.
When the moon is below the sun's altitude the moon is dark and a New Moon occurs.
When the moon is above the altitude of the sun the moon is fully lit and a Full Moon occurs.
The time between two full moons, or between successive occurrences of the same phase, is about 29.53 days (29 days, 12 hours, 44 minutes) on average. This denotes the cycle of alternating altitudes.
Why does everyone see the same phase?
Q: Why does everyone on earth see the same phase of the moon?
A: They don't. The phase you see varies depending on your location on earth. In FET this is explained by the different observers standing on either side of the moon. On one side it is right-side up, and on the other side it is upside down.
Imagine a green arrow suspended horizontally above your head pointing to the North. Standing 50 feet to the South of the arrow it is pointing "downwards" towards the Northern horizon. Standing 50 feet to the North of the arrow, looking back at it, it points "upwards" above your head to the North. The arrow flip-flops, pointing down or away from the horizon depending on which side you stand.
The lunar phase varies depending on where you stand on a Round Earth as well. Here is the RET explanation for why the moon turns upside down when you stand on either side of it: http://web.archive.org/web/20070218184023/http://www.seed.slb.com/qa2/FAQView.cfm?ID=1137