Moon calendar, or lunar calendar, is a calendar in which days are numbered within moon phase cycles. A moon phase is the appearance of the part of the Moon which is currently illuminated by the Sun, as observed from the Earth. It changes cyclically, depending both on the Moon's position in orbit around the Earth and the Earth's position in orbit around the Sun. The 'full moon' phase appears when the Sun and Moon are on the opposite sides of the Earth and the 'new moon' ('dark moon') phase when they are on the same side. The time between two full moons is about 29 and a half days on average, and is called a 'lunar month' or 'lunation'.
An example of the moon calendar is the Islamic calendar, which is mainly used for religious purposes. A year in Islamic calendar consists of 12 months which are not synced with the seasons, for they drift each solar year by 11-12 days. There are presumably other moon calendar representations, some of which are believed to be dating back as far as 25,000 BC.
Most lunar calendars are actually lunisolar calendars, which means that months reflect the lunar cycle, but also include leap months ('intercalary' months) that are added for the purpose of synchronising the calendar year with the solar year. Examples include the Chinese and Hindu calendars, as well as many calendar systems used in the ancient times. All lunisolar calendars have a variable number of months in a year. This is because a year is not evenly divisible by an exact number of lunations, so without the addition of leap months the seasons would drift each year. The result of this is a thirteen-month year every two or three years. Some lunar calendars are synchronised by annual natural events. These are affected by lunar cycles as well as the solar cycle.
Moon calendars are used for determining traditional holidays in certain parts of the world, such as Japan, China, Vietnam, Nepal and Korea. They differ as to which day is the first day of the month -- for example, in Chinese calendar, the first day of a month is the day when an astronomical new moon occurs in the particular time zone. In other moon calendars a month may begin on the day after the full moon or the new moon, etc.