The two most common psychological approaches to literature are Freudian and Jungian, based on the theories of Sigmund Freud and Carl Jung. There is some overlap between their theories, and in fact, until he broke from Freud, Jung was Freud's disciple.
One aspect of Freudian literary analysis involves the concepts of the ego, the id, and the superego, and I remember seeing a wonderful lesson plan on the website of the National Council of Teachers of English that involved applying those concepts to the characters in The Cat in the Hat. Of course, these concepts can be applied to a character in many literary texts. For example, it might be interesting to discuss how Scout in To Kill a Mockingbird manages to integrate all three into a cohesive self over the course of the novel.
Jungian analysis uses concepts like archetypes, which are based on types from mythology and fairy tales. Jung believed that we all have a "collective unconscious," which caused us to act out myths and stories according to the archetypes we identified with. I should point out that this is a gross simplification of Jung's idea, and it would be a good idea to learn more about his theories, which are quite interesting, particularly in a literary context. Robertson Davies, a Canadian author, wrote novels rich in Jungian archetypes. The Deptford Trilogy is one that is particularly easy to apply Jungian analysis to.