The Oni (鬼?) was a cartoonish, goofy looking pipsqueak that wasn't very bright. He was imprisoned in a box 5,000 years ago by a monkey priest, but the box rotted, allowing him to escape. He went around possessing Kuno, Ryoga and others to do supposedly "evil" deeds. The only way to exorcise him was to hit the possessed. Once he possessed Kasumi, nobody wanted to hit her. He was released by a sutra the Oni Temple priest had.
In the anime, during the time he spent inside several people, he absorbed evil energy of those and revealed its true form which was that of a hideous, powerful, demon. After fighting Ranma and Ryoga, Happosai returned from his recent raid. The Oni sensed his immense amount of evil energy, and possessed him hoping to absorb his power. But it struggled with Happosai's own powerful evil mind. Ranma took this opportunity to seal them both in a new box, with some help from Ryoga, and they threw it in the river, hoping never to see it again. But the next day, a news broadcast from America reported two demons terrorizing the shores of California, which were in fact Happosai and the Oni. In the manga he is very weak and stupid, more of a nuissance than an actual threat.
Oni are beasts from traditional Japanese folklore; powerful, magically-talented ogres with one to three horns on their head clothed in tiger skins.