Hyper Sonic is a form similar to Super Sonic in that it makes Sonic faster, invincible, requires fifty rings and makes Sonic glow. It also saps away rings at the same rate of one per second.
However, in addition to requiring the seven Chaos Emeralds it requires the seven Super Emeralds from Sonic 3 & Knuckles which are obtained by entering a large ring with all seven Chaos Emeralds. Sonic is then warped to the Hidden Palace, where he is required to complete one of the Special Stages, which are the same Special Stages that you would get the regular Chaos Emeralds from in Sonic & Knuckles alone. Sonic glows the colors of the Chaos Emeralds in this state, can perform a "double jump" of sorts that can be aimed, and destroys any Badniks on screen. He also cannot drown, unlike Super Sonic who can.
This character was only available in Sonic 3 & Knuckles and has not been in any game since. In the Archie comics, Sonic has a form called "Ultra Sonic"; since this is a common misnomer for Hyper Sonic, and since it appears at the same time as Hyper Tails, it may have been based on Hyper Sonic. Moreover, the Final Color Blaster in the DS version of Sonic Colors holds several similarities to Hyper Sonic. Instead of the Wii version's powered-up Homing Attack, this version's Final Color Blaster involves Sonic becoming enveloped in a rainbow aura, dashing towards the Nega-Wisp Armor, and blasting it to smithereens. Additionally, a deleted voice clip from Sonic Colors describes the Wisps as being "more powerful than Chaos", indicating that Sonic's new powered-up state exceeds the power of the Chaos Emeralds, just like Hyper Sonic.