Phases of the moon

The new moon (when you can’t see it) occurs when the moon is positioned between the earth and sun. The three objects are in alignment. The entire illuminated part of the moon is on the back side of the moon, the half that we cannot see.

At a full moon, the earth, moon, and sun are in alignment, just like the new moon, but the moon is on the opposite side of the earth, so the entire sunlit part of the moon is facing us. The shadowed portion is entirely hidden from view.

The first quarter and third quarter moons (both often called a "half moon"), happen when the moon is at a 90 degree angle compared to the earth and sun. So we are seeing exactly half of the moon illuminated and half in shadow.