The Special World is an area that houses special rooms, which contains important items that are helpful to one's journey. They could be found in multiple areas such in the Fourth Dimension, Super Mario Star Road, or the Star World.
The Special World originates from the 1990-1991 SNES Platformer, Super Mario World (the fourth installment to the Super Mario Bros. series). In the game, the Special World is a post-game bonus rewarded to the player after completing the Star World. It is infamous for featuring some incredibly difficult levels.