Humor Sigmund Freud first proposed the idea that we have various conflicting portions of the mind that are very different in their drives and indeed their "personalities." This was directly tied in with his idea of the unconscious mind as a seperate entity whose thoughts and reactions the conscious mind was unaware of. Freud called his three parts of the personality the Id, the Ego and the Superego. The Ego is the moderator and holds most of the primary personality traits; the Superego is the conscience, mostly occupied with making moral judgements; and the Id is the most primitive part of the mind, existing completely in the unconscious, and seeking solely to gratify its own needs. He conceived of the amoral Id and the moralizing Superego being constantly at war for the attention of the 'main self,' the Ego (like a cartoon with an angel on one shoulder and a devil on the other).
Like many of Freud's other theories, the idea of the unconscious mind as a separate entity has little solid evidence to support it; however, in his idea of the three-part personality, he recognized that even the most single of singlets has different aspects to themselves which can behave like different people under different circumstances.
Carl Jung, one of Freud's contemporaries, went in a slightly different direction. Jung's theories were largely founded on the idea that every human being had certain archetypes existing in their unconscious, and that these archetypes were the same in every person and formed the basis of mythology. His takes on personality included the concept of the "Anima," the feminine aspect existing in all men, and the "Animus," the masculine aspect existing in all women, as well as the "Shadow," which, somewhat like Freud's concept of the id, contained all the 'forbidden' and repressed desires of the conscious mind.
Other psychologists have recognized that each "normal" person seems to be an internal community. See self-pluralism for example. In transactional analysis, the patient is encouraged to temporarily view the different parts of their personality as different persons representing various archetypes, such as "child" and "parent." The concept of the "inner child," so popular in the 80s, was a takeoff on this. More recently, psychologist John Rowan proposed the concept of "subpersonalities" existing within every self. Even among singlets, it's common to hear "I'm a different person at work than I am with my family."
Outside of Western culture, other societies have their own concept of the different 'personae' existing within every self. A cornerstone of Japanese society is the idea that every person has a 'public face' and a 'private face,' which can differ markedly from each other in their behavior and responses.
However, it still seems clear that there is a distinct difference between 'different parts of the personality' and 'different self,' even if many singlets find it beneficial to temporarily view aspects of their personality as being seperate individuals.