From Sonic Retro
The Chaos Rings are six powerful rings that appear in Chaotix. They were formed in the dimensional space created by the Newtrogic High Zone's ancient Special Rings, where Master Emerald Pillar energy crystallized into six rings.
Throughout the game, the Chaos Rings are collected in much the same way the Chaos Emeralds are in the original Sonic the Hedgehog. If the player completes a Level with 50 or more Rings a Special Ring will appear. Jumping into the Special Ring will transport the player to the Special Stage, which, if successfully completed, will award the player a Chaos Ring.
Collecting all six is the only way to thwart Eggman's ambitions and attain the "good ending", with cameo appearances from Sonic and Tails.
If the player completes the game with fewer than six Chaos Rings, then any Chaos Rings they have collected will be shattered by the power of Dr. Eggman's giant Dark Ring. The "bad ending" will then play over the credits, with a transformed Metal Sonic menacing a burning city.
The Chaos Ring storyline goes completely unmentioned in the US manual, which features a drastically different plot, revolving around Dr. Robotnik trying to steal the "Power Emerald" from "Carnival Island". The Chaos Rings can still be collected, but are never explained.
During development these rings were known as the Holy Rings. They were renamed "Chaos Rings" by Chaotix prototype 0202. It is unknown why their names were changed, though it may have something to do with religious censorship employed by many video game publishers at the time.