A lunar year or lunar calendar is one that is based on the cycles of the moon phases. The problem with a lunar calendar is that it drifts away from the seasons. Each year, the start and end dates of each month drift by 11 days. In order to stay correct, every lunar calendar has to deal with this drift away from the calendar year.
Let’s examine a year. A lunar month lasts 29.53 days. So after 12 lunar months, you’re about about 354 days. This is short of the 365 days that it takes the Earth to orbit the Sun. This is a problem since after about 3 years, the lunar months are out of cycle with the solar year by about a month. And this problem would just continue.
To make the lunar calendar work in China, farmers would add in a leap month every 3 years. This would mostly get the lunar month to line up with the solar year, but they still drifted apart somewhat. For some calendars used for religious purposes, such as the Islamic Hirji calendar, they never bothered to sync up the calendars and let them drift. It takes 33 years for the cycle of lunar years to get back to the original position.
A lunar calendar was used in England up until Tudor times.
You can listen to a very interesting podcast about the formation of the Moon from Astronomy Cast, Episode 17: Where Did the Moon Come From?