The answer
The Sun always lights one half of the Moon, and as the Moon orbits Earth we see more or less of that lit half.
The Moon makes no light of its own. It shines because sunlight bounces off it, and at any moment the Sun lights exactly half of it, the half facing the Sun.
The Moon goes around Earth about once a month. When it's between Earth and the Sun, its lit half faces away from us and we see almost nothing: a new moon.
As it moves around, we start to see a sliver of the lit half (a crescent), then half of the face we see (a first quarter), then more (gibbous), until it's on the far side of Earth from the Sun and we see the whole lit face: a full moon. Then it all happens in reverse.
From one full moon to the next takes about 29.5 days. That cycle is where the idea of a month came from.
Go deeper
Why? Why isn't there an eclipse every month?
The Moon's orbit is tilted about 5 degrees compared with Earth's orbit around the Sun. So most months the Moon passes a little above or below the line between Earth and the Sun, and the shadows miss.
Why? Why do we always see the same side of the Moon?
The Moon spins exactly once for every trip around Earth, because Earth's gravity slowed its spin long ago. This is called tidal locking, and it keeps the same face pointed at us.
Why? Can you see the Moon in the daytime?
Yes, often. Except near new moon, the Moon is up in the daytime for part of every day. It's bright enough to see against the blue sky.
Keep wondering
Sources
- Moon Phases, NASA Science
- Earth's Moon: Facts, NASA Science