While the answer by @dwjohnston is good, it is not entirely accurate.
The term transgender covers the wider range of people who do not fit the cis-gender binary of just "male" and "female". This can include people who are intersex, transsexual, gender-queer and agender. Please note that these are all adjectives. A transgender person is not "a transgender".
The term transsexual literally means "to change sex from one to another". Someone may be a transsexual without having sex reassignment surgery (also knows as gender confirmation surgery) to change their primary sex characteristics, but may be on a regime of hormones to change the secondary sex characteristics to align their body more closely with their identified gender. This term applies to both transsexual men and women.
Not every transsexual person needs or requires surgery. The correct terms are:
- Non-operative transsexual: Someone who has or is using hormone replacement (such as oestrogen for trans women or testosterone for trans men), but does not feel the need to have sex reassignment surgery.
- Pre-operative transsexual: Someone who requires surgery to feel complete but has not yet had the procedure.
- Post-operative transsexual: Someone who has had sex reassignment or gender confirmation surgery.
Again, all of these terms apply to trans men and trans women. For more information, refer to the World Professional Association for Transgender Health Standards of Care V7.
Of course, if you are wondering what you should address someone as who is transgender, the rule of thumb is to address them as the gender they are presenting as, or what term/pronouns they ask you use.