Your sex is what you are defined as in terms of your genetics and your biological makeup, for example if you have two X chromosomes, you are of the female sex. Gender, however, is how you identify. For example, you can have an XY chomosome pair (which is the marker for the male sex) but identify as a woman. This is an example of being transgender, trans- simply meaning different. If you are transgender, you identify as a gender different to the sex you were born as. If you identify as the same gender as the sex you were born as, you’re technically cisgender (cis- meaning the same).
Someone who is transgender identifies as a different gender to their birth sex, this is often found in ‘drag’ culture. Whereas someone who is a transexual has had a ‘gender reassignment’ so that they outwardly appear to be of the opposite sex in a biological way. For example, someone who is a transexual* may have had surgery and hormone therapy so that they feel they can live as the opposite sex, the sex they felt they should’ve been born as. The term ‘transvestite’ is pretty much transgender and is fairly outdated by ever-changing, modern-day standards.
Transgender is to do with gender identity, transexual is to do with biological sex: trans 101.
LGBT (or LGBTQ) standards for Lesbian, Gay, Bisexual and Transgender (Q = community). Although being transgender or transexual is not sexuality, it is to do with gender identity. Many people received prejudice and discrimination over the expression of their gender and sexual preferences; the first step to fighting this is through awareness. Being transgender is the feeling that your mind is of the opposite sex to your body (in gender-binary terms**), simple as.
*’Transexual’ is the word often used but the technical term using rules of the English language is transsexual with two Ss, but this is actually quite trivial.
**’Gender binary’ refers to the idea of the existence of only two unique genders; female (XX sex chromosome pair) and male (XY sex chromosome pair). Some people prefer an identity that does not neatly fall into one category or the other, even if they may have the reproductive organs of only one biological sex. Some people are born with abnormalities on the chromosome pair that determines sex, resulting in combinations such as X (Turner’s Syndrome), XXX (Triple-X Syndrome)and XXY (Klinefelter Syndrome); these genetic conditions are largely irrelevant to transsexualism.