Hey, Bradley here.
Someone who is "masculine" is expected to be dominant, prone to rational thought (as opposed to emotional), strong, powerful, in control, authoritative... etc.
On the other hand, someone who is "feminine" is expected to be gentle, submissive, emotional and compassionate, nurturing, delicate, beautiful... etc.
However, the human mind has both masculine and feminine tendencies. I think what causes problems is when men are hyper-masculinized and women are hyper-feminized. Just look down the aisles of a toy store to see the polarization: the boys aisle is full of robots, trucks, things that shoot, or that transform into something. This is perhaps the case because little boys are supposedly more excited by movement. The girls' aisle is full of dolls, castles, stickers, furry things, and things to play dress-up with. This is because little girls supposedly are more excited by how things look, or what colors they are (as opposed to how they move). With such polarization of gender identities especially in children, it's no wonder the boys and girls rarely play together. I mean, it's not as though little Jimmy would want to be caught dead playing at Sally's house with her pink 8-ball and princess castle. Nor would Sally want to be caught dead playing with Jimmy's build-your-own-robot kit and transformers. So boys play with boys, girls play with girls, and there is little room for anyone who does not quite fit either category.
Even into adulthood, we still have this hyper-masculinization and hyper-feminization of the gender roles. So when someone feels they don't fit their gender role the way they're "supposed to", they feel like they have to go to the other extreme - the gays are expected to behave like straight women, and the lesbians are expected to behave like straight men, which causes ALL KINDS of stereotyping problems. Because, clearly not all gay men fit that stereotype, and likewise, not all lesbian women fit their stereotype, (and the bisexuals don't fit EITHER category, haha) and the whole thing is one big tangled mess.
This is why the LGBT community is seen as so strange by the rest of society - because the L's G's B's and T's don't fit cleanly into the boxes labeled masculine and feminine, or even the boxes labeled boy and girl. Then, all the in-between people are seen as outcasts.
And don't even get me started on gender roles in the context of a relationship - why does the guy have to "be the guy" and why does the girl have to "be the girl", and when the couple has two men or two women, why does one of the girls have to "be the guy" or one of the guys have to "be the girl" in the relationship. The roles of dominant and submissive do NOT have to be clearly assigned. The whole thing is just absolutely absurd!
So to recap - sex refers to a person's biology (which is by no means fixed), and gender refers to the way society expects that person to behave, dress, express themselves (which aught to be much more fluid than it is).
So the next time someone asks you "are you a boy or a girl", or even "are you THE boy or THE girl", you can explain all about the difference between sex and gender.