According to the fifth edition of the Diagnostic and Statistical Manual of Mental Disorders (DSM-5), Gender Dysphoria is the “official diagnosis” for people whose gender at birth is contrary to the one they identify with. In order to be diagnosed as such, however, not only must the individual identify with another gender, but there must be significant distress caused by not being the gender they most closely relate to, and the distress must last for longer than 6 consecutive months. In children, the desire must be present and the child must verbalize such.
Gender Dysphoria is manifested in various way, including but not limited to: strong desires to be treated as the opposite sex or be rid of their current sex characteristics (ie genitalia and/or breasts or lack thereof), or the existence of a strong conviction that one displays feelings and/or reactions that are typical of the sex with which they identify.
The biggest improvement to the DSM-5 from previous editions is the change in phrasing from “Gender Identity Disorder” which suggests that it is unnatural or makes one ill/crazy to be transgender. Calling it a dysphoria (defined as a profound state of uneasiness or dissatisfaction) validates it as a legitimate feeling or way to identify as opposed to an illness that needs to be “cured”.