Psychotherapy has many different disciplines, and so has many different methods of achieving the same goal. Briefly, all psychotherapies strive to improve the mental state of the patient through personal counseling, which allows the patient to confront and accept or reject personal issues which might be unconsciously affecting their mental state.
Psychoanalysis was first practiced by Sigmund Freud, who is considered the creator of modern psychological therapeutic techniques. It is still practiced today, but has undergone many changes by other practitioners. The basis of psychoanalysis is verbalization of unconscious thought, with the understanding that much formative material in the mind is not considered consciously, but instead sits under the surface and affects thought-processes. Psychoanalysis seeks to bring harmful or suppressed memory to the surface to be confronted and accepted by the patient. There are over twenty different practices of psychoanalysis, so methods and goals differ widely.
Person-centered therapy is a more recent form of psychoanalysis created by Carl Rogers, who sought to give the patient a comfortable place where personal defenses can be dropped. Often, patients with mental issues create conscious or unconscious walls between themselves and the outside world in order to keep from being hurt or exposing themselves to others, and the therapist attempts to give them a safe haven so they can "be themselves" without risk of judgement or ridicule. This type of therapy is very popular, as it can be practiced with a minimum of training and does not require great knowledge of psychology, instead relying on the ability of the therapist to empathize with the patient. While Psychoanalysis can be confrontational, Person-centered therapy seeks to be empathic.
Behavior therapy is a method of Conditioning, and was first described in the United States by B.F. Skinner, who theorized that behavior could be altered with reinforcement, either negative or positive. The aim of behavior therapy is to encourage proper behavior (thought or action) while discouraging improper behavior. This can take many different therapeutic forms, from Systematic Desensitization (exposing an arachnophobe to spiders until the fear is dulled by familiarity), to Observational Learning (watching "proper" behavior and imitating it), to Habit Reversal Training (taking one aspect of life and seeking to deliberately alter it through controlled planning and learning). Behavior therapy is more hands-on and intrusive, and can be strenuous to patients in a fragile mental state.