Welcome Edge

The second question is easier but not necessarily simple. Feeling like both male and female, but not in between, is called "bigender" by some. From what I've read, it seems to mean the person experiences being two genders at the same time, or else at different times or situations.
Androgyne, on the other hand, is generally meant as a single gender per person that is "between" male and female, or spans across male and female.
These aren't exact terms though and different people will use them differently.
As for the first question... I don't think anyone really knows what gender is. It's like consciousness. It's a thing that does not really exist in a measurable way, but has almost universal importance.
Gender is two things, at the very least:
It's the set of rules society has -- whether they are laws, expected behavior, or stereotypical rules of thumb. It is pervasive and insidious in our society. Every time someone says "ladies and gentlemen" or calls a cat "she" by default, they are reinforcing ideas about gender. Every time they wear clothes or take a step or speak or act in any way, they are expressing gender and performing gender, even if in very subtle unconscious ways.
Gender is also what we feel and what we do in relation to those rules. And in some ways I think we are oversimplifying ourselves by it.