How do the moon phases happen?

Answer 1

The phase of the moon (new, half, full, etc) is a function of how much of its lit surface we can see from the earth. Half the moon is always sunlit, half is dark, but as the moon orbits the earth, more or less of the lit face is pointed towards us here on earth. The moon's synodic period around the earth is about 29.5 days, which coincides with the length of a month. Once in each of its synodic periods, its lit face is directly oriented towards earth -- a full moon. Because the synodic period is less than some calendar months, on relatively rare occasions we can have two full moons within one month. The second of those full moons is referred to as a "blue moon" -- thus the phrase, once in a blue moon, which means "rarely."

Answer 2

The moon has phases because of its relative position to the Earth and the Sun. As the moon orbits the Earth, sunlight hits different parts of its surface, causing different portions to be illuminated or shadowed from our perspective on Earth. This creates the varying moon phases we observe.