Transgender is an umbrella word that is used to describe a very large and diverse group of people. "Transgender" can refer to transsexual people; to genderqueer and gender-variant people; to crossdressers; even to feminine men who still call themselves men, and masculine women who still call themselves women.
If someone says "I am a woman," or "I am a man," or "I am ____," please take that person seriously. Our cultural framework tends to tell us that their bodies may contradict their statements — that there's no way you could be a guy with XX chromosomes, or a genderless person with an obvious beard. But the trans person is the one who's right, and the simplistic framework is the model that's wrong. Gender is not dependent on physical appearances, or on the word of doctors, friends, family. The individuals are the ones who get to assert their own identity.