The definition of transgender is someone who identifies as something different than what they were born as. Non-binary people are also transgender because they identify as something different than what they are born as. Not all transgender people are transexual though.
What is being transsexual?
A transexual person is a person who wants to change the sexual characteristics of their body. For instance, if you are a trans girl who wants to go on HRT (hormone replacement therapy), you would be considered a transexual person. Not all transgender people are transexual. Transexual as a term has fallen out of use but it still does have some use whenever we are talking about the differences in transgender people and the different ways they might want to transition. As a warning, some people consider the word transphobic on the basis that it alienates out people who are transgender. For example, someone who would say “I’m not transgender, I’m transexual” could be taken as being transphobic since it is pushing away people, usually non-binary people, making it seem like they are crazy or outlandish. Implying you don’t want to be included with them.
What being transgender is not:
Being transgender is NOT the same as being a crossdresser or a transvestite. Crossdresser is defined as someone, usually men, who dresses up in the opposite sexes clothing. A transvestite is the same idea as a crossdresser but someone who dresses up in the opposite sexes clothing for purely sexual or arousal reasons. A trans person doesn’t choose to transition just for sexual reasons, your transition of course will have some sexual aspect of it because you are a human being after all. However, people don’t transition just solely for that reason. People commonly confuse crossdressers and transgender people because they don’t know the difference. Transphobia is almost always rooted in ignorance after all. Another important thing to add to is crossdresser and transvestites don’t experience gender dysphoria, while trans people do.
To finish, transgender people are a very unique group that commonly gets confused with other groups, such as crossdressers or transvestites. There is absolutely nothing wrong with being either a crossdresser or a transvestite, but they are not transgender. When people make the mistake of mixing those groups together it further pushes transphobia, for example seeing a trans woman and thinking she is a cross-dressing man in a dress is transphobic. Trans women are women and trans men are men. Being able to recognize that what you were doing before was wrong and growing upon it is how you become a better supporter and a better person.