Gender may have always been presented to you as an either/or choice, two options, boy/girl, man/woman. But that's not really the way gender works. There are many different genders, and over time people have developed just as many terms to describe their experience of gender.
The term transsexual usually refers to someone whose gender identity is different from the gender they were called at birth and who lives currently with (or in, or as, different words could be used to reflect different people's experience) a gender that is different from the one they were assigned when they were born. So someone who was called a boy when they were born, but has always felt themselves to be a girl and a woman, who identifies and lives their adult live as a woman, might identify themselves as transsexual, or as a transsexual woman.
This doesn't mean that you are either a woman or a man or transsexual. Gender categories aren't always so discrete. Rather one way to think about it is that transsexual refers to the experience of crossing through or over gender identities, or of transitioning from one gender to another gender.
Most of the time people who identify as transsexual change parts of their body and their appearance in order to present and look like how they feel. This might include surgery or taking hormones. Not everyone who identifies as transsexual does this. There's no "test" or requirements people must pass in order to be transsexual (although the medical establishment has something different to say about that).
However this is one of the things that usually distinguishes the term transsexual from the term transgender, which is more broad and includes people who may or may not be living as a gender different than they were given at birth and who often have not chosen to change their bodies in way to permanently alter aspects of how they move through the world.
Just as there are many ways people experience and express gender, there are many ways people describe who they are, and terminology evolves. Trying to know the "right" word to use is difficult because what fits for one person may not for another.
So when it comes to gender definitions, from a sex education perspective, what seems best is to follow other people's leads. To not assume you know how a person identifies, and if you use one term and they ask you to use another to acknowledge their request and respect it.
It can also help to get other people's perspectives on language, particularly people who use the words you are interested in knowing more about. Here is some suggested reading:
TransYouth Family Allies - Learning the Lingo
Gender Spectrum - Gender Terminologies
Survivor Project - Guide to Intersex and Trans Terminologies