Like the Earth itself, the Moon is roughly spherical in shape. But instead of orbiting the Sun (once per year), it orbits the Earth instead (roughly once per month).
At any given point in its orbit, one side of the Moon is facing the sun and is lit up, while the other side faces away and is dark. But because it's orbiting overhead, we see different sides of it:
- When it's directly opposite the sun, we only get to look at the bright side, so we see it as a full moon
- When it's on the same side as the sun, we're facing the dark side, so we see a new moon. Or actually, we don't see anything at all, since it's dark and it only comes out during the day.
- When it's halfway between those, we see it as half-full. (These are called "quarters", I guess because they're a quarter of the way through the cycle.)
All the other phases are just steps in between these.