The gem that was the “Heart of the Mountain” in The Hobbit is the Arkenstone.
The Arkenstone is the most treasured heirloom of Thorin Oakenshield's royal family. It is a large, luminous white jewel discovered by the dwarves beneath the Lonely Mountain and cherished by the rulers of Erebor. Thorin calls it the Heart of the Mountain because it represents both the kingdom's heritage and his claim to its throne.
Bilbo finds the Arkenstone while exploring Smaug's treasure hoard after the dragon has been driven from Erebor. He secretly keeps it, recognizing that it could become a bargaining tool when Thorin refuses to negotiate with the Men of Lake-town and the Elves.
The stone is not the One Ring, mithril, or a Silmaril. Those are also important objects in Tolkien's legendarium, but they belong to different stories and have different histories. Bilbo eventually gives the Arkenstone to the besiegers, hoping to prevent a battle, which enrages Thorin before the Battle of Five Armies.