The ‘Metonic Cycle’ is a particular approximation of relationship between the solar year and the lunar month. The ‘Callipic Cycle’ is an even better approximation.
The solar year, or ‘Tropical Year’ is the length of time that the Sun, as viewed from the Earth, takes to return to the same position, or equinox, in the sky. The ‘Sideral Month’, is the length of time that the Moon, as viewed from the Earth, takes to return to the same position relative to the stars on the celestial sphere. However, because both the orbit of the Moon around the Earth, and the Earth around the Sun are completely independent, this leads to some complications.
The cause of the phases of the moon is that from the Earth we see the part of the Moon that is illuminated by the Sun from different angles as the Moon traverses its orbit. So the appearance depends on the position of the Moon with respect to the Sun (as seen from the Earth). Because the Earth orbits the Sun, it takes the Moon extra time (after completing a Sidereal Month, i.e. a full circle) to catch up and return to the same position with respect to the Sun. This longer period is called the ‘Synodic Month’ or lunar month
A ‘year’ of 12 Synodic months is about 354 days on average, 11 days short of the 365 day solar year. So about every 3 years an extra 13th synodic month has to be inserted into the calendar. The Greek astronomer Meton of Athens introduced a formula for the relationship between Synodic months and the Topical year in 432 BC. It was his observation that a period of 19 solar years is almost exactly 235 lunar months, and rounded to full days equates 6940 days.
In the following century Callippus developed the Callippic cycle. This was a more accurate approximation, obtained by taking one day away from every fourth of Meton's cycles, so creating a 76-year cycle with a mean year of exactly 365.25 days.
The Metonic and Callipic cycles are key to modeling a long term lunar / solar calendar, but the 19 year cycle is also close to an eclipse cycle, and could potentially predict eclipses for about 4 or 5 recurrences of eclipses. However, the Ancient Greeks had knowledge of better eclipse cycles.
Further reading: [bib]487[/bib], [bib]489[/bib]