FYI, a transexual/ transgender person is someone that identifies (psychologically) as a member of the opposite biological (genetic) sex. Typically these people develop normal sexual organs and there is no specific medical (biological) cause for this condition, which can occur at anytime. Some individuals state that they have always felt that they are the "wrong sex", while others develop these feelings at puberty and later.
Generally these people also consider themselves to be heterosexuals, EG., a transgender man (biological woman) would be attracted to women.
Now a hermaphrodite is something usually different. A hermaphrodite is someone that has both male and female reproductive organs. As far as we know there has never been a human hermaphrodite, as there is no record of a person having both testicular and ovarian cells at the same time.
The term is commonly used to refer to a condition known as "ambiguous genitalia" and falls under the umbrella of what has been accepted as the condition of intersexuality. This covers several different types of conditions caused by hormones and other genetic factors during fetal development that results in deformation of internal or external sexual organs.
Intersexuality also covers people that have the opposite sexual organs of their biological sex (not to be confused with transgender), a rare condition that does occur, which can be whole or partial- such as a man (XY chromosomes) with a penis, testicles and a uterus, or a woman (XX chromosomes) with a penis, testicles and no uterus or ovaries, etc. Remember sex is based on chromosomes and not sexual organs.
Another condition is a person that is both male and female, having both sets of chromosomes, which is very rare, but does happen. This person may develop any of the previously mentioned conditions, but they usually favor the development of female sex organs since they are the ones that first occur in he fetus.