Something very basic and very useful when talking about these things is to keep in mind the difference between sex and gender. They're used synonymously in most everyday conversation, but people who talk about these things a lot generally use sex to refer to anatomy or genetics, and gender to refer to a person's mental or social identity. (For example, having a vagina and XX chromosomes are characteristics of female sex, while wearing a dress and calling onesself Mrs. instead of Mr. are characteristics of female gender.)

With those definitions in place, it's a lot easier to describe what "transsexual" means. In a nutshell, a transsexual is a person whose sex does not match their gender. A longer but more exact definition would be something like "a person who was born with a male-sexed body and a female-gendered mind, or a female-sexed body and a male-gendered mind". The various medical treatments we TS's tend to undergo - hormone therapy, electrolysis to remove hair, surgery to alter genitalia or to add or remove breasts, etc. - all have the goal of modifying a person's sex to be in line with their gender.

(Of course, no amount of surgery or medication can alter a person's chromosomes, and every transsexual goes through some period where they identify as one gender but out of necessity live as the other in most parts of their life, until it becomes feasible to "come out" as their true gender. In other words, it's possible for someone to be male-sexed in some ways and female-sexed in others thanks to incomplete medical treatments, and since "gender" can refer to personal identity but also to social identity it's possible to be male-gendered in one way but female-gendered in others, too. That's why the "sex doesn't match their gender" definition isn't as exact as it could be.)