As you might guess, Quarter Days are a British and Irish tradition that take place every three months throughout the year—well, almost every three months. A couple days ago we covered equinoxes and solstices, and the quarter days approximately match those four events.
Christmas on 25 December is one, and Lady Day on 25 March is another. The summer solstice, 24 June (of course technically the 24th is a few days after the solstice, for reasons explained here) was called Midsummer Day, but also for the Roman Catholic Church it was the Feast of the nativity of John the Baptist.
The last one would be the autumnal equinox, 24 September, but the Church already had a day at the end of September, the Feast of Michael and All Angels, also called Michaelmas. Instead of celebrating two big days so close together, the fourth Quarter Day was the 29th of September. Just as Lady Day was the start of contracts with tenant farmers, Michaelmas was the end of the harvest season, when accounts would be settled and tithes from the harvest collected. In Ireland, Michaelmas was a day the rents were due.
There were also cross-quarter days, called so because they fell between the Quarter Days. They were four other holidays that existed before Christianity came to the British isles. They should get their own post, and they will...tomorrow.