There are six Chaos Emeralds hidden throughout South Island. Collecting all of them is necessary to achieve the 'good' ending of the game. To access a Special Stage, you must reach the end of Act 1 or Act 2 of any Zone with 50 or more rings. At the goal post, a giant ring will appear above it. Jump into this giant ring to be transported to a Special Stage.
Each Special Stage is a rotating maze where Sonic is constantly moving. Your goal is to navigate the maze, avoid the 'GOAL' blocks (which end the stage prematurely), and collect the Chaos Emerald. There are also 'bumper' blocks that bounce Sonic around. The key is to learn the layout of each stage and anticipate Sonic's movements. You can control Sonic's direction slightly with the D-Pad, but his momentum is largely fixed. Practice is essential, as these stages can be quite challenging, especially the later ones. If you hit a 'GOAL' block or run out of time, you'll be returned to the main game without the Emerald.