It does not have a name but it is usually referred to as the dark side of the Moon (This doesn't mean it has no light) or the far side of the Moon.
quarter Moon if its half of a sphere then its half moon
A horizontal half moon is called a crescent moon.
when the moon half shows itself, it's called a half moon.
It is called "first quarter" or "last quarter" when the moon is exactly half-lit.
The official name for it is actually Luna, but you very rarely hear it being called that.
The final half of the moon cycle is called the waning phase, while the first half is called the waxing phase. During the waxing phase, the illuminated portion of the moon visible from Earth is increasing, whereas during the waning phase, it is decreasing.
A lunar eclipse can be photographed during the night when the moon is visible and the eclipse is occurring.
The moon would be called a waxing gibbous when it is just larger than a half.
That phase is called the third quarter or it could be called a waning half moon.
The Half Moon was Henry Hudson's ship.
The half moon is also called the "first quarter" or "third quarter" moon, depending on its phase. In the first quarter, the right half of the moon is illuminated, while in the third quarter, the left half is lit. This terminology refers to the moon's position in relation to the Earth and the Sun.
When the right half of the moon is illuminated it would be called first quarter. (That's when you see the Moon from the northern hemisphere.)