Transsexual people experience a gender identity inconsistent or not culturally associated with their assigned sex, i.e. in which a person’s assigned sex at birth conflicts with their psychological gender. A medical diagnosis of gender dysphoria can be made if a person expresses a desire to live and be accepted as a member of their identified sex, or if a person experiences impaired functioning or distress as a result of their gender identification.
Transsexual people are stigmatized in many parts of the world, though some cultures have historically embraced gender diverse people. Transsexual people have become more widely known in Western culture since the mid to late 20th century, concurrently with the sexual revolution and the development of sex reassignment surgery and cross-sex hormones (HRT – Hormone Replacement Therapy).
It is worth noting that in scenarios other than those related to medicine, the using the word “transexual”is not generally accepted. As being transgender isn’t a sexuality and isn’t related to those, it is generally considered a slur.
More information about this disorder can be found at WikiPedia.