All right, I'll try to explain this whole sex/gender business in (what I hope will be seen as) simple terms. Only problem being that I'm Swedish, so there is a small risk I may not be using the English terms the way they should be (although I don't think I'm screwing up).
SexThis is your physical sex, the one your doctor said when you were born. It's based on your genitals and secondary sexual characteristics. It's a very physical thing and it's actually not as simple as male/female - there's tons of variation even in sex. The important part is that it is purely biological and about your body.
Gender (identity)Your self-identified gender identity is the gender you feel you belong to. For instance, if you naturally feel you should raise your hand when you hear "all boys, raise your hand" then you're male-gendered. If you raise your hand when the teacher asks about girls, then you're female-gendered. If you don't feel inclined to raise in either situation, then you're probably non-gender or identify along a third gender (many cultures recognise genders aside from simply male and female - western culture is at heart ignorant about these identities). And who knows? Perhaps you didn't even see why you couldn't be both or more.
This is a bit of oversimplification, since many feel forced by society to raise their hand in a situation like this according to society's demand rather than their own. This pressure may be so strong they do not notice their own self-identified gender for many, many years. If we had a more tolerant society, this scenario would likely work just fine. The important part is that gender identity does not necessarily co-relate to how your body looks like at all. Even though certain combinations (female-gendered, female-sexed) are more common.
(Gender) expressionWhat do you feel like wearing? What kind of body language do you feel comes natural? Perhaps you think lipstick looks gorgeous, and perhaps you think that rugged look where you haven't shaved for two days (but not three!) is optimal. In what tones of voice do you speak? Etc.
Your gender expression does not necessarily need to match your gender at all. Our culture has done a damn good job of telling people you can only do certain things depending on what's between your legs and even how you identify, but you can do what you please. You could be a male-sexed, male-gendered person who likes doing nail polish. Hey, nail polish is pretty freaking rad - have you seen all the ways you can express yourself with it? Perhaps another person is female-sexed, female-gendered and wants to wear mostly trousers. This has gotten a lot more accepted during the last hundred years, but it wasn't that long ago trousers were only acceptable on men in our culture.
As you may notice, wearing trousers is not inherently male or female. Our cultural norms dictate that tons of expression of voice, clothing or whatnot
is only for one gender or the other though - cause as a society we're still bigoted like that. It's all about how one chooses to express one's own identity, gendered or no.
EDIT: I originally planned for this post to make a difference between sex and legal sex, but that's a system that may be inherently Swedish. I've heard most people aren't given a social security number that mentions sex outside of my country, hence it's irrelevant for most people. I changed the text a bit in removing traces of legal sex existing and hope those who read the post's first draft didn't feel confused