You're thinking of a transsexual. Transgender means a person's gender doesn't coincide with their biological sex. That's the only real definition.
So if you're male, with a penis and XY chromosomes, but your brain tells you you're a woman? Presto. You are trans. If you're female, with a vagina and XX chromosomes, but your brain tells you you're a man? Boom. You're trans, too.
There are people out there (more than you might realize) who are chromosomally (and, often, genitally) neither male nor female at birth.
Lots of trans people, historically, tend to keep to themselves. You'll never know they're one or the other because they're no more vocal about their gender than you or I are.