Gender dysphoria is a condition where a person experiences discomfort or distress because there's a mismatch between their biological sex and gender identity. It's sometimes known as gender identity disorder (GID), gender incongruence or transgenderism.
Biological sex is assigned at birth, depending on the appearance of the genitals. Gender identity is the gender that a person "identifies" with or feels themselves to be.
While biological sex and gender identity are the same for most people, this isn't the case for everyone. For example, some people may have the anatomy of a man, but identify themselves as a woman, while others may not feel they're definitively either male or female.
This mismatch between sex and gender identity can lead to distressing and uncomfortable feelings that are called gender dysphoria. Gender dysphoria is a recognised medical condition, for which treatment is sometimes appropriate. It's not a mental illness.
Some people with gender dysphoria have a strong and persistent desire to live according to their gender identity, rather than their biological sex. These people are sometimes called transsexual or trans people. Some trans people have treatment to make their physical appearance more consistent with their gender identity.