What is this thing?
Gender, despite efforts to conflate it with sex, means the presentation and behavior associated with one's sex. Society expects men to be masculine and women to be feminine, although what constitutes "masculine" and "feminine" varies from culture to culture. There is no hard and fast link between femaleness and femininity or between maleness and masculinity, but people can still be rather rigid in their expectations.
Dysphoria is the opposite of euphoria. It comes from Greek roots meaning "difficult to bear." Euphoria is being on a high. Dysphoria is being on a low.
So gender dysphoria would be dissatisfaction with one's gender, or perhaps with the gendered behavior and presentation that someone feels are expected of them. A woman for whom stereotypical femininity is not a good fit might experience gender dysphoria. Same for a man who dislikes stereotypical masculinity.
I find it not just puzzling but actually misleading when people use gender dysphoria as a synonym for transsexualism. Dissatisfaction with gender expectations is not at all the same as having a birth defect in which the sex of your brain is opposite that of your body. People who transition to female may or may not be stereotypically feminine. People who transition to male may or may not be stereotypically masculine. People born transsexual change anatomical sex. It doesn't require hormones or surgery to alleviate dysphoria with your gender.
People born transsexual still get saddled with a diagnosis of Gender Identity Disorder, which is also a misnomer. We submit to such nonsense so we can meet our actual need, which is to change sex. One hopes that someday the medical and psychological establishments will finally get it.