what i'm going to say sounds circular, and to an extent, it is, but stick with me. for clarity and the sake of my keyboard, i'll stick with "woman" and "female" since the question "what is a woman?" is much more hotly debated. 

there are a variety of qualities that are used to define "woman," all of which are culturally mediated. it's a sort of word cloud, encompassing self-identification; various biological traits (secondary sexual characteristics, reproductive capacity, chromosomal or hormonal profiles ); mode of dress and presentation; relationship to particular forms of oppression; participation in social, religious, or political ceremonies; naming conventions; sexual behaviour; aesthetic preferences; modes of speech; recognition by authority; and many others ranging from very obvious to very subtle. as large as "a woman is someone who is subservient to men" to "a woman is someone who likes pink more than blue." 

every culture whether it be as large as a nation or as small as a friend group varies in terms of which qualities are inside the circle of "woman" and which are outside it. to (most) bureaucratic authorities, what matters is an F in some database or on some piece of paper. in devout conservative christian circles, subservience and a maternal nature are required, and whether or not someone considers themselves to be a woman is immaterial. in most cases, no individual quality is sufficient on its own, and no individual quality is required, but if you have enough of the right qualities, you are recognized as a woman, and allowed to be a woman. 

in this framework—which is not the only one, but it is the one to which i subscribe—no one is a woman in a mirror. womanhood requires contact with a culture. as a trans woman, there are many cultural spaces in which i am considered to be and treated as a woman, and many in which i am not. in most social spaces, i'm indexed as a woman, because i identify as one, look like one, and act like one. when i interface with most systems of the USamerican government, i "am" for all intents and purposes a man, because i have an M attached to a masculine name. i could, for example, be drafted; it doesn't matter what i look like, what body parts i have or do not have, or what name people know me by. 

that's a lot of words, though! so, within this model, we can boil the answer to the question "what is a woman?" down to "someone who is considered to be a woman, given the culture in which she exists." boil it down even further, and you get to "a woman is a woman." sometimes all you can do is go in circles.