Dark Yuugi (also known as Yami no Yuugi, or Yami Yuugi and known as Yami or the Pharaoh in the dub) is the spirit that resides in the Millennium Puzzle that is worn by series protagonist Yuugi Mutou. He is very strong and confident, and usually takes control of Yuugi's body whenever Yuugi or his friends are in danger. Yuugi typically relinquishes control to Dark Yuugi during duels, though during some critical duels (vs. Pegasus J. Crawford, vs. Possessed Jounouchi) Yuugi has maintained or taken control during a duel.
In the first half of the series (both anime and manga), Yuugi calls Yami no Yuugi "Mou hitori no boku" (literally, "the other me"), while the others, once they figure out that Yami no Yuugi is different from Yuugi call him "Mou hitori no Yuugi" ("the other Yuugi"). It took quite a long time before anyone else recognized Dark Yuugi as a separate entity or soul, and even Jounouchi is prone to tripping up and calling him "Yuugi" at times. However, until the final arc of the series, Dark Yuugi had no name nor memory of his past, so he didn't mind being called Yuugi.
In the first arc of the manga and in the Toei animated series, Dark Yuugi took control when Yuugi was weak or fell unconscious. After he had punished those that had harmed Yuugi or his friends, Dark Yuugi retreated into Yuugi's now-fragmented soul, within the Millennium Puzzle. Yuugi often had no memory of anything that happened when Dark Yuugi took control. In the manga and the Toei anime, Yuugi started to remember things shortly before Seto Kaiba's Death-T project. In the Yu-Gi-Oh! Duel Monsters anime, it is more unclear when Yuugi became truly aware of Dark Yuugi's presence, though he had come to accept it by the end of Duelist Kingdom.
A common misconception is that Dark Yuugi is physically different from Yuugi, but the truth is that he is not. He is not actually taller than Yuugi, nor are his eyes different shape. However, it can be said that his eyes somewhat change color (from Yuugi's purple shade to more of a red-violet), and Yuugi's bangs stand up, creating the appearance of "lightning spikes" that stand up whenever Dark Yuugi is in control. He also stands with more confidence than Yuugi, giving him the illusion of being taller. Anzu Mazaki also thought that Dark Yuugi had a deeper voice than Yuugi, which would contribute to her crush on Yuugi's darker half, before she knew that they were one and the same person (or rather, two different souls residing in the same body). Despite this, even Yuugi's closest friends couldn't always tell Yuugi and Dark Yuugi apart.
Dark Yuugi's archrival is Seto Kaiba, a rivalry that is believed to have stemmed all the way back from Ancient Egypt. In the Battle City arc, the Tombkeeper Isis Ishtar shows both Dark Yuugi and Seto Kaiba a tablet from thousands of years ago that shows two "duelists" with marked resemblances to both Kaiba and Dark Yuugi battling one another with their signature monsters, the Blue-Eyes White Dragon and Black Magician, respectively. Since then, Dark Yuugi has come to accept that he has no memories of his true/previous life due to something that occurred thousands of years ago in Ancient Egypt, and that his spirit is that of a long-dead Pharaoh.
Spoiler warning: Plot and/or ending details follow.
In the final arc of both the anime and the manga, he loses the title of "Nameless Pharaoh" and discovers his true name from the past: Atem. In the very last part of the series Dark Yuugi and Yuugi duel each other in a "ceremonial battle" that will determine if Dark Yuugi is separated from Yuugi and returns to the place where he belongs.