half moon
That phase is called the third quarter or it could be called a waning half moon.
This is the first quarter phase. The moon looks half lit, half in shadow, but you actually can only see one quarter of the moons surface at this point (during a full moon, you would only be able to see half of the moons surface - the half that faces us).
That's called the "Third Quarter" phase. It occurs about 22 days after the New Moon.
crescent moon
First Quarter and/or Third Quarter.
Those shapes, and the corresponding phases, are called "crescents" ... when the lighted part that we see is less than half of a full disk.
"Waxing" is the term used when the Moon's illuminated part is growing in size, while "waning" means that the lighted part is decreasing. "Gibbous" means more than half, but not full. So "waxing gibbous" is the phase of the Moon between the first quarter and the full moon.
A waxing gibbous moon appears high in the east at sunset. It's more than half-lighted, but less than full.
Gibbous phase is from full to half.
That would be Third Quarter.
Last quarter.
The lighted half of the moon is always facing the sun. The entire dark half of the moon is facing earth at the time of "New Moon".