Ask most people what a blue moon is and they will say the second full moon in a calendar month. That definition is now universal, and it is about eighty years old — the result of an error that nobody caught for decades.
The original definition
The Maine Farmers’ Almanac used a seasonal rule. Each season normally has three full moons, and each has a traditional name tied to its position in the season. Occasionally a season contains four. In that case the third of the four — not the last — was called the blue moon, a placeholder that kept the remaining names correctly aligned with the season.
It is a strange rule, and the almanac never explained it clearly, which is precisely how the trouble started.
The 1946 error
In March 1946, Sky & Telescope published an article by James Hugh Pruett attempting to explain the almanac’s rule. Pruett worked from a 1937 almanac entry, misread the seasonal system, and concluded that a blue moon was simply the second full moon in a month.
The definition sat quietly in the literature until it was used on a radio programme in 1980 and then in a widely sold board game, after which it spread everywhere. Sky & Telescope published a correction in 1999, tracing its own mistake — by which point the wrong definition had thoroughly won.
Which should you use
The calendar-month definition, because that is what everyone means. This site uses it, and flags a blue moon whenever a second full moon falls inside one calendar month. It happens roughly every two and a half years, which is where the phrase “once in a blue moon” gets its sense of infrequency.
One quirk worth noting: because the definition depends on calendar months and clock time, a blue moon is time-zone dependent. A full moon at 00:30 UTC on the 1st falls on the previous month in the Americas, so a month can contain a blue moon in London and not in New York.
Actually blue moons
The moon can genuinely appear blue, though it has nothing to do with the calendar. It requires atmospheric particles of a particular size — around one micron — which scatter red light preferentially and let blue through. Large forest fires and volcanic eruptions can produce them. After Krakatoa erupted in 1883 the moon reportedly appeared blue or green for roughly two years.