As everyone has already mentioned, there’s a lot of debate on this topic. Among academics, it’s not unusual for people to talk about gender (and the binary male/female categories of gender) as being completely a social construct (though not, obviously, a weak one). In general I think that it’s understood that a lot of the role of being one gender or another is learned. After all, think how different gender roles are from culture to culture (though I suppose it’s hard to see this right off without first hand experience).
On the other hand, there are a couple of things that seem to be inherent differences between the sexes in general. Men produce more testosterone, for example—an order of magnitude more—which is accepted as causing more aggressive behavior. I seem to remember there being one other trait that everyone accepted as being an inherent difference between genders. But basically everything else is up for grabs right now, in that it hasn’t been credibly shown to be genetic. (There’s a great piece on This American Life about testosterone, by the way. thisamericanlife.org)
Also, there have been some studies that seem to indicate that men and women’s brains function differently (men’s brains are bigger, but women’s function more efficiently, with more connections). However, the outcome of what they can do is essentially the same.
Personally, I think that it’s worth remembering that there is a huge range and variation inside each gender. I think there may be some not yet understood biological factors, but the way they manifest is highly unpredictable. Also, there is clearly a performative quality to certain visions of womanhood and of manhood that is culturally dependent and learned. But this too varies radically. So we all have a chance to be people, if we’re lucky, and not just men or women.