Sonic the Hedgehog 4: Episode I

To get a Chaos Emerald, you need to clear a Special Stage. In order to get to a Special Stage, you need to clear any non-boss level with 50 or more rings. As you run past the sign at the end of the level, you'll see a giant floating ring. Just jump into it before the level ends, and you'll get to play the Special Stage! Once in the Special Stage, the three main obstacles to getting the Chaos Emerald are as follows:
Time
Rings
Exit Spheres

Time is constantly ticking down, so you have to move quickly. There are bubbles you can nab to add a few seconds to your time, so keep an eye out for them.
Some gates that you need to open require that you have a certain number of rings, so grab as many as you can!
Exit Spheres are the red circles with "!" on them. Avoid these at all costs, as hitting them will instantly exit you from the Special Stage.

As an additional pointer, if you're nearly out of time, you can pause and go to Retry. It will start the Special Stage over with no penalty, and you won't have to run through a level with 50 rings again.

See my solution for The Story Begins for how to unlock this and the Ring Collector achievement: