Makai (魔界) is the world of demons and basically means "world of spirits, hell." The entrance is located in a cave in the mountains near the Hakurei Shrine.
This is where the heroines first met Alice Margatroid, apparently before she lived in the Forest of Magic. It is ruled over by Shinki who created Makai.
This is where many events of Highly Responsive to Prayers occurred, as well as the entirety of Mystic Square and the last two stages of Undefined Fantastic Object.