Gender dysphoria refers to the distress someone experiences when they have a disconnection between their bodily sex and internal sense of gender. The diagnostic label gender identity disorder (GID) was used by the Diagnostic and Statistical Manual of Mental Disorders (DSM) until its reclassification as gender dysphoria in 2013, with the release of the DSM-5.1
Transgender can refer to a man who identifies as a woman, or a woman who identifies as a man. Some activists go so far as to say that a “trans woman” (that is, a man who identifies as woman) is a woman.
Not all people who suffer from gender dysphoria identify as transgender and not everyone who identifies as transgender suffers from gender dysphoria.
1. See Anderson, When Harry Became Sally, pp. 28, 94-95, on the evolution of the DSM.