As Dr. Mazeroff expressed in his presentation, the psychoanalysis of fairy tales developed many years after the origins of fairy tales themselves. There are two main psychologists who were highly influential in the development psychoanalysis of fairy tales, Sigmund Freud and Carl Jung.
According to Freud, the human psyche has three primary components, the id, the ego, and the superego. The id being the primal, base instincts, the superego being the repressive , controlling part, and the ego being the balance of the two. These components appear in many of the classic fairy tales. One prime example is Peter Pan, where Peter, who is always free and does what he wants, represents the id, and Hook, who hates Peter for how free he is, represents the superego. Wendy and Smee, who are the balances of Peter and Hook, forming some balance between fun freedom and control, represent the ego. Similarly, in Little Red Riding Hood, the mother who warns Riding Hood not to go flirting with strangers, is the superego. The wolf, with his lure and primal lust, is the id, and Riding Hood who must choose between the two, the ego.
Jung on the other hand, puts the human psyche into four different categories. The anima, the animal, the self and the shadow. The anima and animal are similar to Freud's id, and represent the animal-like, primal instincts of a person, with the anima representing the feminine portion, and the animal representing the masculine person. The self represents the good aspects of a person, who they want to be, and the shadow the negative aspects that people try to fight in themselves. This model, too, is found in many fairy tales. For example, in Hansel and Gretel, the children represent the animal and anima, as they are craving primal needs such as food. In the story they not only eat their fill of the ginger bread house, but continue to engorge themselves on the house. Hansel, being male and taking initiative and showing ambition, is the masculine animal, and Gretel, who is female, submissive and good at house work, is the feminine anima. The father and mother, who are supposed to be kind and loving, if not slightly corrupted, are the self, while the step-mother and witch, who are evil and have notable bad qualities represent the shadow.
I personally find Jung's model to be more accurate at analyzing fairy tales than Freud's. Particularly, gender is very important in fairy tales, and males and females act very differently. Jung's separation of anima and animal, while I don't find it particularly true in real life, is a very accurate analysis of fairy tales.