I figured I would start with the basics. Here is a site that can help.
Here is how I explain it. You feel like you are born in the wrong body. Your entire life (usually it is your entire life) you feel like you were meant to be the other gender/sex. They grow up hoping they will magically change into that suddenly. It is NOT the same as being homosexual. Homosexuality is about sexual orientation and which sex you are attracted to. Transgender is about what sex/gender you think you really are.
However, transgender isn't black or white either. You can be male to female (MTF), female to male (FTM), male to neutral/genderqueer/androgynous (MTN) or female to neutral/genderqueer/androgynous (FTN). If there are others I missed I am sorry. There are some who wish to remain neutral on the issue and don't want to pass as either gender/sex. Also, many transgender individuals have moments where they want to act and dress like the sex/gender they were born into.