The harvest moon is the full moon closest to the autumnal equinox. In the Northern Hemisphere, this usually happens in September, though it may occasionally happen in October. In the Southern Hemisphere, this takes place in March or April.
Throughout the year, the moon rises at a different time each day. Every moonrise is usually about an hour later than the previous moonrise and the moonrise times cycle throughout the hours of the day all month to rise right around the time of the sunset when the moon is full (near sunrise on the new moon) and then an hour later the day after the full moon, etc.
However, at the time of the harvest moon, the angle of the ecliptic, the moon’s orbital path, is shortened. So, instead of rising an hour later, it rises about a half-hour later for a few days, giving you the day-lengthening effect of a full moon for 3 or so days instead of just one.
The large, bright moon rising just as the sun sets several days in a row effectively extends the light hours of the day which is helpful for people hurrying to get their crops in before winter. Thus this moon is called the Harvest Moon.