Great question, and one that's pretty fraught at the moment. I don't have an answer beyond a tautology - a man is someone who identifies as a man - and the knowledge that some cultures assign adherence to certain behavioural norms to that (ex. A man acts as breadwinner, is competitive, has a certain type of physicality distinct from women, etc.), most of which crumble with any hard look at them.

To be frank, I don't really care about what a man or woman is. If identifying as a man if female, or a woman if male, makes it so someone doesn't want to blow their brains out, then that's a cool and good thing. But note the distinction - man != male and woman != female in my statement.