Lunar phases are different stages in the cycle of the moon's orbit around the Earth. As the moon moves, people on Earth see varying levels of its illuminated surface, from a new moon, in which the moon can't be seen at all, to a full moon, in which people see the entire portion of the moon illuminated by the Sun. This cycle takes a little less than 30 days to complete, and it may be known collectively as the lunar cycle.
People have been tracking lunar phases for thousands of years. The obvious waxing and waning of the moon was clearly visible to human societies, and many cultures based their measurements of months around the moon's movements. They also associated particular phases with various events and beliefs. For example, some cultures believed that waning moons were associated with bad outcomes in new endeavors.
The lunar phases are fairly easily explained by astronomy. At any given time, fully half of the moon and half of the Earth are illuminated by sunlight. As the moon orbits around the Earth, varying levels of the illuminated area can be seen by people standing on Earth. When the moon is between the Earth and the Sun, the illuminated side faces away, creating a new moon. Full moons occur in the opposite situation, when the Earth is between the moon and the Sun.
One might expect a constantly cycle of lunar and solar eclipses which would correspond with the lunar phases, except that the orbits of the Earth and moon are angled, rather than occurring along a straight line. This means that the perfect alignment needed for an eclipse is actually fairly rare.
When the moon is growing, it is said to be “waxing,” while a “waning” moon is shrinking. In the Northern Hemisphere, the phases of the moon move from the right to the left side of the moon's surface: if only the right quarter is illuminated, the moon is waxing towards full, while a small sliver of darkness on the right side of the moon indicates that the moon is on the wane. The opposite is true in the Southern Hemisphere, and along the equator, the crescents of waning and waxing moons actually appear in a horizontal alignment.
The variation in the appearance of the lunar phases around the world is explained by the relationship between the angles of the moon, Earth, and Sun. Changes in angle will alter the way people see the moon, depending on where they are. These variations also explain why sunrise and sunset occur at different times.