Transgender: someone whose gender identity is different than their biological sex.
Transsexual: term used to describe a transgender person who is looking to or has changed their biological sex to match their gender identity.
Transwoman: someone who is or was biologically male and identifies as female.
Transman: someone who is or was biologically female and identifies as male.
Transvestite or crossdresser: a person who dresses or performs the mannerisms of a sex or gender that is different than their own.
Genderqueer: someone who rejects typical binary systems of sex or gender and therefore create their own gender identity.
Cisgender: term for someone whose gender matches their biological sex. So for example: I am cisgender. I was born female and I identify as female.