The older definition, which is recorded in early issues of the Maine Farmer's Almanac, states that the blue moon is the third full moon in a season that has four full moons. Why would one want to identify the third full moon in a season of four full moons? The answer is complex, and has to do with the Christian ecclesiastical calendar.
Some years have an extra full moon—13 instead of 12. Since the identity of the moons was important in the ecclesiastical calendar (the Paschal Moon, for example, used to be crucial for determining the date of Easter), a year with a 13th moon skewed the calendar, since there were names for only 12 moons. By identifying the extra, 13th moon as a blue moon, the ecclesiastical calendar was able to stay on track