Multiple Personality Disorder is a psychiatric diagnosis. It describes a condition in which a person displays multiple distinct identities or personalities (known as alter egos or alters), each with its own pattern of perceiving and interacting with the environment. Each alternate personality has a set of memories, behaviors, thoughts, and emotions unique to that personality.
The disorder was the focus of the episode "City Sushi", wherein Butters Stotch is erroneously treated for the disorder and discovers his therapist Dr. William Janus actually does suffer from the condition. After being blamed for the actions of one of Janus' alternate personalities, Butters exposes him to the town. Janus is then permanently fixed to a single personality, that of the City Wok owner.
While Janus is the only confirmed instance, Herbert Garrison displayed extremely strong dissociative behavior with his puppet Mr. Hat (and briefly Mr. Twig), due to his repressed homosexuality. These patterns generally subsided once he came out as gay, leading to Mr. Hat being discarded, though in his only appearance since, Garrison seems to lose control of him and again treat him as a separate identity.
Although not specifically referred to as such, Eric Cartman also displays numerous worrying symptoms resembling multiple personality disorder: he has numerous alternate personalities, foremost amongst which is Mitch Conner (his left hand), which appears to act independently of Cartman on multiple occasions (examples include the episodes "Fat Butt and Pancake Head", "200", and "201").
Several more alternate personalities emerged during the episode "1%", in which Cartman projected certain aspects of his personality onto his stuffed toys (including Clyde Frog, Polly Prissypants, Muscleman Marc, Rumpertumskin, and Peter Panda). He genuinely appeared to view these toys as separate individuals, conversing with them and vocalizing their roles. His behavior ultimately extended to his acting without his own knowledge and against his own will.
It is likely that Cartman's conflicting interests of wishing to both "grow up" and hold on to his stuffed toys caused him to externalize the conflicting elements of his personality onto his toys, with Polly Prissypants in particular embodying his wish to grow up.
Although so rare in real-life that many doubt its existence, multiple personality disorder is a common subject in popular culture. The real disorder is conventionally referred to as dissociative identity disorder, but South Park presents it under the more recognizable name.