The revolution of the Moon around the Earth makes the Moon appear as if it is changing shape in the sky. This is caused by the different angles from which we see the bright part of the Moon's surface.
These are called "phases" of the Moon. Of course, the Moon doesn't generate any light itself; it just reflects the light of the Sun. The Moon passes through four major shapes during a cycle that repeats itself every 29.5 days. The phases always follow one another in the same order.
Today's Moon Phase
by the U.S. Naval Observatory
The phases were understood as early as 500 BC by the Greeks. The key clue is that the phase of the Moon correlates with its angular distance from the Sun. For instance, the Moon is in its crescent phase when it is near the Sun in the sky but full when it is opposite the Sun. The Greeks realized this implies the Moon is a solid sphere, in orbit about the Earth, half of which is always illuminated by the Sun.
In the figure above you are looking down on the Earth's North Pole. The Earth spins counterclockwise (in 24 hours), and the Moon orbits counterclockwise (in 27 days). The fraction of the Moon's sunlit hemisphere which we can see from Earth determines the lunar phase at any time. We see a "full," "crescent," or dark ("new") Moon depending on the angle between the Sun and Moon as viewed from Earth.
New Moon
This is the first of the moon phases
Waxing Crescent Moon
"Waxing" means growing and refers to
the size of the illuminated part of the moon that is increasing.
First Quarter Moon
Waxing Gibbous Moon
Full Moon
The full moon occurs when the Moon lies on the opposite side of Earth from the Sun. The moon as seen from the surface of the earth is fully illuminated by the sun at this time, presenting a "full" round disc to viewers on earth. As always, only half the total surface of the moon is illuminated.
The full moon reaches its highest elevation at midnight
A full moon is the only time when a lunar eclipse is possible; at that time the moon may move through the shadow cast by the earth. However, because of the tilt of the moon's orbit around the earth relative to the earth's orbit around the sun, the moon may pass above or below the shadow, so a lunar eclipse does not occur at every full moon.
Full moons are generally a poor time to conduct astronomical observations, since the bright reflected sunlight from the moon overwhelms the dimmer light from stars.
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".