I find some of that "gender is social" to be baffling and a little sexist. Now I have zero problem with transpeople, it's their life and their choices but when someone tries to oversell it to me that e is a woman because she wears dresses and has huge fake boobs, what about those girls that don't identify as dudes but prefer to dress more masculine and don't wear jewels? Gender is to me just something biological, it shouldn't define your interests or roles in society or aspirations, but maybe they just have explained the whole thing wrong to me.
Wikipedia actually explains it fairly well:
http://en.wikipedia.org/wiki/Sex_and_gender_distinction
Generally speaking, the biological term is 'sex'. Even sex isn't very clear: there's intersex people, underdevelopment of sexual characteristics, hormonal differences etc. It's not all that clear-cut. But if you want to refer to a person's biological state of being, that's the term you'd use. With trans people it's generally very offensive to do so, though.
Gender generally doesn't refer to a person's biology, but to the social/cultural context of a person's...sex? So for instance, I could be biologically female but my gender could be male because I conform to those gender roles (dress, fake boobs, the way I act etc).
Generally speaking, though, it's polite to let people define their own gender and refer to them however they want to be referred to. And there's really no good reason not to.