From Sonic Retro
The Super Emeralds are a second set of 7 emeralds introduced in Sonic the Hedgehog 3 & Knuckles. They are not found in either Sonic the Hedgehog 3 or Sonic & Knuckles alone.
Sonic the Hedgehog 3 & Knuckles
To obtain the Super Emeralds, the player must first find and touch a flashing, multicolored Super Ring. The first Super Ring is found at the beginning of Mushroom Hill Zone Act 1, in an area not accessible in the stand-alone version of Sonic & Knuckles. Further Super Rings only appear if the player has collected all seven Chaos Emeralds, after which they will replace Special Rings.
Super Rings transport the character to the Hidden Palace, where any Chaos Emeralds the player has collected are transformed into into large, gray emeralds. By giving up any Chaos Emeralds, the player will lose the ability to transform into Super Sonic or Super Knuckles. Jumping onto one of the dull gray emeralds whisks the player away into a Special Stage, identical to those in Sonic & Knuckles. If the player successfully completes a Special Stage, they will be awarded a Super Emerald.
Collecting all 7 Super Emeralds grants the player the ability to transform into Hyper Sonic, Hyper Knuckles, or Super Tails.
The seven Super Emeralds, along with the Master Emerald, in the Emerald Shrine located in Hidden Palace Zone.
The remains of the Super Emeralds make an appearance in their resting place in Hidden Palace, reached at the end of Lava Reef Zone Act 2 as Knuckles. Unfortunately, they have all turned gray and cracked, and are now completely drained of their power and thus can no longer be used. According to Christian Whitehead, these stone relics were created when the Chaos Emeralds once again drifted off into the Special Stage dimension.
In other media
Sonic the Hedgehog (Archie comics)
The Super Emeralds were introduced in Archie's Sonic the Hedgehog comic series with issue #71. With the power of a single Super Emerald, Sonic transforms into Super-Sonic [sic], Ultra-Sonic, Solar-Sonic, Polar-Sonic and Eco-Sonic, each transformation influenced by the environment he runs through. More Super Emeralds were introduced in later issues.