It's a good question, Mrsfrumble. Female is the accurate term for a mammal with the potential to produce young (has internal gonads.) Male is the mammal with the potential to inseminate, and has external gonads.
Woman and man are the respective terms for humans.
When speaking of gender, we should use feminine and masculine. But those two words have acquired specific overtones that don't always apply to what we mean by gender. So people tend to say 'female gender' - which conflates gender with sex.
Gender's a great big, overwhelming, conglomeration of restrictions and expectations - which is tied to one's sex, like it or not. As a woman you can crop your hair and be a bricklayer, swig pints and be a Millwall fan, and you'll still be perceived as female (which you are) and will still suffer gendered oppression of one sort and another.
I think the current fashion for young adults to label themselves with a glittering array of genders & sexualities might be an effort to break down the strictures of gender. It won't work, though, because they're still accepting gender as a fact. 18 is too young to understand how massive it is and how subtle, especially as not many 18-year-olds have been exposed to any proper analysis or critique of gender. Few of them are even relating it to physical sex whereas, in practice, it's undeniably linked.
... This hasn't come out quite right. Hope someone else has posted a more cogent summary!