Definitions that refer to gender are almost always more complicated than you first imagine. This is in part because most of us are raised to think that gender is a binary, you are either one thing or another. In reality, most of us experience gender as something more complicated than that, even if we don't ever talk about it with others.
Transgender is a broad or umbrella term that refers to people whose gender identity in some way is not the same as the gender they were identified as at birth. This may or may not mean that a person chooses to live as one particular gender, or as a gender that is different than what they were called when they were born. Instead it describes an experience or reality of not matching up neatly with the thing you were first called.
Lots of people use the term transgender to describe either their gender identity or their experience of gender, including people who are androgynous, transsexual, cross-dressers, and folks who in other ways don't conform to the gender binary.
What usually distinguishes the term transgender from transsexual is, first that it encompasses a wider range of gender expression, and second, that people who identity as transgender may not want to transition fully to some other gender.
This is a very general definition and it's important to remember that no definition is going to fit for everyone. If you are interested in learning about terminology because you want to be able to use language that will open up conversations rather than shut them down, it's important to read many different perspectives, and particularly perspectives from people who use the word transgender to identify themselves. There are hundreds of places to find these definitions online, here are a few places to start:
TransYouth Family Allies - Learning the Lingo
Gender Spectrum - Gender Terminologies
Survivor Project - Guide to Intersex and Trans Terminologies