Most calendars are in some way based on either the solar (Sun) or the lunar (Moon) cycle; some are based on both. Of the commonly used calendars:* The Gregorian calendar - the most widely used calendar worldwide - is based on the Sun. That is, the length of the year is based on the Sun; but the start of the year and the months is not related to any particular astronomical event.
* The Jewish calendar is based on both the Moon and the Sun. As far as I know, it's the only widely used calendar that is based on BOTH. Some years have 12 months, others have 13 months.
* The Muslim calendar is based on the Moon. A muslim year is about 11 days shorter than a Gregorian (solar) year.
* Many other calendars are based on the Sun.