A transsexual (sometimes spelt transexual) person is someone who establishes a permanent identity with the opposite gender to their assigned (usually at birth) sex. Transsexual men and women make or desire to make a transition from their birth sex to that of the opposite sex. This requires adoption of all the behaviours of the opposite sex, especially dress. There is normally some type of medical alteration (gender reassignment therapy) to their body.
The stereotypical explanation is of a "woman trapped in a man's body" or vice versa, although many in the transsexual community reject this formulation.