Lunar phases are the result of our seeing the illuminated half of the Moon at different angles. The Moon exhibits different phases as the relative positions of the Sun, Earth and Moon change, appearing as the full moon when the Sun and Moon are on opposite sides of the Earth, and becoming invisible as the new moon when they are on the same side. The time between two full moons is 29.5 days; it is longer than the time it takes the Moon to orbit the Earth since the Earth-Moon system is orbiting the Sun. The phases are not created by the shadow of the Earth on the moon; instead, they are a result of our seeing only part of the illuminated half of the Moon.
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.
Note that the plane of the Moon's orbit around the Earth is tilted by about 5 degrees with respect to the Earth's orbit around the Sun. Therefore, eclipses of the Moon during the full Moon and of the Earth by a new Moon are rare and usually newsworthy.
In the Northern hemisphere, if the right side of the Moon is dark, the light part is shrinking: the Moon is waning (moving towards a new Moon). If the left side is dark, the Moon is waxing (moving towards a full Moon). The acronym mnemonic "DOC" represents this ("D" is the waxing Moon; "O" the full moon; and "C" the waning moon). In the Southern hemisphere, this is reversed, and the mnemonic is "COD". A french mnemonic is that the waxing moon at its first "premier" quarter phase looks like a 'p', and the waning moon at its last "dernier" quarter looks like a 'd'. One more (Northern hemisphere) mnemonic, which works for most Romance languages, says that the Moon is a liar: it spells "C", as in crescere (Italian for "to grow") when it wanes, and "D" as in decrescere ("decrease") when it waxes.