A blue moon is an extra full moon that shows up in a subdivision of the year: either the third or a second full moon in a month of the common calendar.
The phrase has nothing to do with the actual colour of the moon, even though a literal “blue moon” (the moon showing up with a tinge of blue) may happen in a certain atmospheric condition: e.g., if volcanic eruptions or fires leave particles in the climate of simply the correct size to prudentially scatter red light.
The term has usually referred to an “extra” full moon, where a year which usually has 12 full moons has 13 instead. The “blue moon” reference is connected to the third full moon in a season with four full moons, in this way correcting the timing of the last month of a season that would be generally expected too early. This occur every two to three years (seven times in the Metonic cycle of 19 years).
The March 1946 issue of Sky & Telescope misjudged the traditional definition, which prompted the advanced colloquial misunderstanding that a blue moon is a second full moon in a single solar calendar month with no regular connection. Due to the rarity of a blue moon, the term “blue moon” is utilized informally to mean a rare event, as in the phase “once in a blue moon”.
One lunation (a usual lunar cycle) is 29.53 days. There are about 365.24 days in a year. Therefore, about 12.37 lunations (365.24 days divided by 29.53 days) happen in a tropical year. In the broadly used Gregorian calendar, there are 12 months (the word month derived from moon) in a year, and usually there is one full moon every month. Each calendar year contains approximately 11 days more than the number of days in 12 lunar cycles. The additional days accumulate, so every two or three years (seven times in the 19-year Metonic cycle), there is an additional full moon. The additional full moon essentially falls in one of the four seasons, giving that season for full moons rather than standard three, and, hence, a blue moon.