Dark Bakura is the malevolent spirit trapped inside the Millennium Ring, and he's basically Yami Yugi's arch enemy. He's been holding a grudge for 3,000 years—the two have serious history. His whole goal is to collect all the Millennium Items and open the Door of Darkness, which would unleash evil power. Since he has no body of his own, he hijacks Bakura's, usually without permission.
Early on, Dark Bakura made Bakura's life miserable by taking over whenever he played games with his friends. He'd activate Penalty Games that trapped their souls into Monster World miniatures, forcing Bakura to keep switching schools. Yugi and his friends eventually managed to weaken his grip by beating Dark Master Zorc in that game's final battle.
Things got complicated after that. When the group nearly got trapped in Duelist Kingdom's labyrinth, Dark Bakura conned Bakura into putting the Ring back on, claiming he'd changed and would help save everyone. For a while, it was genuinely unclear what his deal was. He seemed to help out occasionally and gave Bakura more autonomy than before—but that was all part of the manipulation. Behind the scenes, he murdered Pegasus by ripping out the Millennium Eye and taking it for himself. The Dungeon Dice Monsters arc made his true intentions crystal clear: he'd secretly planted a piece of his soul into the Millennium Puzzle to eventually uncover the True Door from inside it. Every helpful move was calculated. He's been waiting for the moment all the Millennium Items come together so he can finally open the Door of Darkness and release what's sealed within. And the whole time, Bakura has no idea his other half keeps taking control whenever it suits him.
Content compiled by AnimeList.moe from publicly available sources.