Sex is your biological status: what organs and hormone levels do you have, and how have they developed? Sounds straight-forward, at least at first.
Gender might be defined as a social role and group identity you take on which is influenced most significantly in most people by their sex. So most people pick from one of the two massively dominant genders, wind up pretty content about it, and have organs matching everyone else in their camp.
But what if you have testes and breasts? And hormone levels pretty much in between the standard man and the standard woman? You might end up legally forced to adopt an 'official' sex based on your chromosome data or what went on your birth certificate, perhaps, but does that help you pick a gender? Does that actually reflect your sex? Probably not. Do you identify more with another sex? What about another gender? If you want to change over, how much will you do and what changes are possible?
The organs you have, the hormone levels you have, and how you feel about them all affect what sex you become and what gender you select. People who aren't comfortable being a traditional man or woman and sleeping with the opposite are simply trying to work out all the permutations and nomenclature now that they're somewhat more free to do so.
If a given person is polite about it and doesn't expect you to memorize a bunch of fluid terms to use for them or coddle their sexuality more than you would anyone else, just let it be and don't worry about the variety of possibilities. They'll work themselves out and they aren't likely to affect you. If they're a dick or an irrational activist about it, and there's plenty of those also, just ignore them and/or fight to keep them from defining your life any more than you're allowed to define theirs.
You can look into the definitions in any Women's or Gender Studies website; or you can ignore it for now and simply be a decent human being to the people you meet, ignoring their chosen combination unless they step on your own rights.