A half moon occurs when only half of the illuminated portion of the moon is visible from Earth. The phase of the moon colloquially known as a half moon is technically referred to as either the first quarter or third quarter phase.
Although the sun illuminates one-half of the sphere, the part of the sphere facing the Earth contains only one-half of the illuminated portion. The other part of the moon that faces Earth is one-half of the dark side of the moon. That side of the moon, which faces away from the sun, doesn't reflect any light and isn't visible to viewers on Earth. While one-half of the moon faces Earth and one-half of the moon is illuminated, only one-quarter of the moon is actually visible on Earth. Flattened into two dimensions, viewers on Earth perceive the illuminated one-quarter of the moon as one-half of a two-dimensional circle, leading to the term half moon.
As the moon cycles from a new moon, the first half moon phase, called the first quarter phase, occurs between the waxing crescent and waxing gibbous phases. After reaching a full moon, the moon begins to wane; the third quarter phase occurs after the waning gibbous phase and before the waning crescent phase.