GLAAD has a good resource on transgender identity: Transgender 101.
Gender identity is someone’s internal, personal sense of being a man or a woman (or as someone outside of that gender binary.) For transgender people, the sex they were assigned at birth and their own internal gender identity do not match.
Trying to change a person’s gender identity is no more successful than trying to change a person’s sexual orientation — it doesn’t work. So most transgender people seek to bring their bodies more into alignment with their gender identity.
People under the transgender umbrella may describe themselves using one (or more) of a wide variety of terms, including transgender, transsexual, and genderqueer. Always use the descriptive term preferred by the individual.
In writing here, I sometimes get tripped up on the differences between sex, gender, and sexual orientation. No more. See also Tips for Allies of Transgender People and An Ally’s Guide to Terminology: Talking About LGBT People & Equality.