Usage is all over the place, which causes some confusion unfortunately. For medical and other precise usage questions, use a reference book. What I can address is the meaning of these terms in the political realm.
Most organizations that once were collectively referred to as gay rights have switched to using the letters GLBT or even longer labels like GLBTIBQQ. The use of the T is for Transgendered not Transsexual.
The reason for this is that Transgendered as a form of address is intended to include all transsexuals as well. It is intended to be a broader term that refers to all sexual minorities who have any discrepancy among the list of traits such as anatomical, chromosomal, self-identification, fashion, etc that makes their gender and/or sex different from what society would usually expect.
So, in summary, my answer is that (at least in political parlance), Transgender is a broad term for at least all of transsexuals AND all cross dressers AND even a few others beyond those two groups.
Intersex people sometimes object to being swept up in such a nebulous catch-all group, so many organizations have added an I for Intersexed as well.
Transsexual: Someone seeking (pre-op) or who has undergone (post-op) procedures (hormones, surgery, etc) to change their anatomical sexual characteristics to better reflect the gender they regard as their true self.
Cross dresser: Someone who expresses, utilizes, or presents as a different gender either in part or in full and either occasionally or routinely, but without seeking to permanently alter their physical sex.
Intersex: Someone whose anatomical or chromosomal sex is either inconsistent or ambiguous. Hermaphrodites are people born with some form of each type of genitalia, for example. Other examples include people with XY chromosomes, but a condition that blocks the ability to for Y chromosomes instructions from being applied, which creates an anatomical female.
