The Emerald Shrine is the resting place of the Master Emerald, and in some cases the Chaos Emeralds or Super Emeralds. It was first depicted in Sonic & Knuckles, as a landmark located in the Hidden Palace.
Hidden Palace Zone
In the final chapter of the Death Egg Saga during Sonic 3 & Knuckles, Sonic and Tails confront Knuckles at the heart of Angel Island: Hidden Palace Zone, next to the Lava Reef volcano. There is a subterranean Emerald Shrine here, with installed on flowery plinths which the Super Emeralds can be placed on and the Master Emerald supported above them. Characters are teleported to this Shrine by hopping into a flashing Giant Ring, which, on the first visit, causes their previously collected Chaos Emeralds to react to the Shrine and place themselves on it, getting upgraded to power-less Super Emeralds in the process, after which the character loses access to his Super form. The player can then fully unlock the emeralds' powers by clearing Special Stages, completing the upgrade process and gaining a more powerful form.
Seen in Sonic Adventure, the Emerald Shrine the Chaos Emeralds were originally discovered in was located outdoors in Mystic Ruins four thousand years ago, next to the Knuckles Clan's city, also housing the Master Emerald. In Tikal's time it also played host to a colony of Chao, and their watery guardian Chaos. Pachacamac's attack on the Shrine led Perfect Chaos to lay waste to the Knuckles Clan. Using a kind of Chaos Control, Tikal sealed Chaos (and herself) inside the Master Emerald to prevent further destruction. No longer having the Chaos Emeralds next to it to cancel its power, the Master Emerald caused half the continent to rise into the sky, Emerald Shrine included. From then on the Shrine existed on Angel Island, and can be visited in the Adventure Field.
Whether this incarnation represents a separate Shrine from the Hidden Palace one, or a retcon, is uncertain.
Cannon's Core and the ARK
In Sonic Adventure 2, Sonic and Knuckles are astounded to discover a replica of the Mystic Ruins altar at the heart of the Space Colony ARK. Constructed by Professor Gerald Robotnik, the imitation Shrine is part of the power supply to the Eclipse Cannon, built in Cannon's Core to allow the weapon to tap the Chaos Emeralds. Biolizard was sealed in the core after the G.U.N. attack 50 years ago, and is fought at the base of the shrine by Sonic and Shadow.
The short boss level at the end of Sonic Advance 3 is Altar Emerald, taking place in what seems to be the shrine seen in Sonic Adventure. Located amidst the vast, cloudy ruins of Chaos Angel, it sees the game's final fight against Eggman, and Gemerl's twisted transformation into the extra final boss. Knuckles pointedly remains here in the game's end sequence, even as the other characters run off to celebrate.
Sonic Chronicles: The Dark Brotherhood
The Nocturnus Echidnas invade Angel Island during Chapter 5 of Sonic Chronicles. Sonic and Knuckles fight their way across the Island in an attempt to stop Imperator Ix accessing the Shrine; in this game, it is located inside a pyramid-temple. Only briefly seen in a cutscene, the Emerald Shrine holds only the Master Emerald.