While the common idea of a “blue moon” is a second full moon in a calendar month, that’s not the original definition. The term arose to address a rare astronomical phenomenon of having four full moons in a season. Seasons are always three months and ordinarily have only three full moons. The third of four upset the natural order and instead of carrying the customary name for that season’s third full moon, it needed to have a different name. For instance, if the Lenten Moon had been the third full moon of four in the winter season, it would have triggered an extra month of fasting before Easter came later than usual in the spring. It was deemed a “betrayer,” for which the old English word was “belewe.” This came to be misconstrued as “blue.” An error in an astronomy magazine in 1946 pegged a second full moon in a calendar month as a blue moon and the notion stuck.
These articles explain blue moons well:
from Smithsonian Magazine
The last time we had a blue moon using the traditional definition was in the spring of 2016, on 5/21/16. The next time it occurs will be in the summer of 2021, on 8/22/21, and after that, the next will be 8/19/24. The average time between these blue moons is 2.7 years (per the Space.com article).
As for a second full moon in a calendar month, that happens in an irregular manner. Recent occurrences were 8/31/12, 7/31/15, and two in one year: 1/31/18 and 3/31/18. The next couple will be 10/31/20 and 8/31/23.
Both types of blue moons are infrequent and noteworthy, but the calendar version is a man-made construct while the third in a season is strictly astronomical.