What is a Moon phase?
Half the Moon is always lit
The Moon makes no light of its own. What we see is sunlight bouncing off its grey rock and dust. At every moment the Sun lights exactly half of the Moon, the half that faces it, just as it lights half of the Earth. The other half is in night.
As the Moon circles us, we see that lit half from a changing angle. When the Moon is on the far side of the Earth from the Sun, its whole day side faces us: a full Moon. When it passes between us and the Sun, its day side faces away and we see almost nothing: a new Moon. In between, we see part of the lit half. That changing shape is the phase.
There are eight named steps. After new Moon comes a waxing crescent, a thin sliver that grows night by night. At first quarter we see half the face lit; it is called a quarter because the Moon is a quarter of the way round. Then comes a waxing gibbous Moon, more than half lit, and the full Moon. After that the same shapes return in reverse, waning gibbous, last quarter and waning crescent, until the Moon slips into the Sun's glare again.