The lunar phase is the portion of the Moon that is visibly illuminated by the Sun as seen from Earth. As the Moon orbits the Earth, the relative positions of the Sun, Earth and Moon change. Since the Moon only appears bright due to the Sun's reflected light, only the half of the Moon closest to the Sun is illuminated.
When the Sun and Moon are on opposite sides of the Earth, the Moon appears full: the moon appears as a whole disc. As the Moon orbits the earth, the moon wanes, as the amount of illuminated lunar surface reduces, until the moon disappears at the New Moon when the Moon is between the Earth and the Sun.
The different phases of the moon have different names. As the moon waxes (the amount of illuminated surface is growing), the moon moves through the New Moon, Crescent Moon, First-Quarter Moon, Gibbous Moon and Full Moon phases, before returning through the Gibbous Moon, Third-quarter Moon, Crescent Moon and Old Moon phases. Old Moon and New Moon are interchangeable, although New Moon is used in preference, and Half Moon is often used to mean the First- and Third-Quarter Moons.
It should be obvious that the Full Moon is highest in the sky at midnight, and that the New Moon is above the horizon during the day.