Calendars based on the cycles of the moon have a shorter year than solar calendars. How that time discrepancy is dealt with depends on the culture.
Ancient cultures typically had two options for creating calendars: solar or lunar. Solar calendars track time based on the movement of the sun in the sky. It takes 365.24 days for the Earth to travel around the sun and make up a year. Lunar calendars however are based on the phases of the moon which restart every 29.5 days adding up to only 354.37 solar days. This leaves an 11 day discrepancy between lunar and solar calendars.
Intercalation of “Lunar” Calendars
To account for this 11 day difference some cultures engage in a practice known as “intercalation” which is the adding of extra days/weeks/months to synchronize your calendar with a solar year of 365.24 days. Many lunar calendars are, in reality, lunisolar calendars as they intercalate extra time to keep their lunar year somewhat aligned to our solar year.
A variety of cultures use lunisolar calendars, especially in East Asia. For example the traditional Chinese calendar is based on lunar cycles but adds a 13th month ever few years. This is why Chinese New Year doesn’t have a fixed date (on our calendar). Intercalation is used to keep the lunar New Year from straying too far which keeps it sometime between late January to late February.
The alternative to adding time is to do nothing about the 11 day discrepancy which has the cumulative effect of pushing holidays further and further around the calendar. This hands off approach can put spring holidays in the fall, winter months in the summer, etc. The Islamic calendar (the Hijri calendar) operates this way, which explains why Muslim religious holidays move around our solar based calendar so much. It takes 33 years for a holiday on a lunar calendar to come back around to its original position.
It’s not just lunisolar calendars that intercalate time. Our calendar year is 365 days but it takes the Earth 365.24 days to travel around the sun. We add time to our Gregorian calendar to account for the extra 0.24 day period of time. We do this by adding a Leap Day every 4 years to even things out.
Added info: The oldest known calendars are a group of carvings from around 32,000 BCE created by the Aurignacian people. These carvings are in antlers, bones, and cave walls found in France which have crescents, dots, and lines diagraming the cycle of the moon. These early lunar calendars document that, for tens of thousands of years, humans have tracked the passage of time by looking to the skies.