I want you to lie for me. If you’re a woman, say “I am a man.” If you’re a man, say “I am a woman.” Say it out loud. Say it to your reflection. Do you feel that little disconnect there, where the sentiment you’re articulating doesn’t match up with the reality you experience? You know you’re lying. Even if someone else comes up and says “Hey! That’s right! That’s definitely what you are,” you will still know you’re lying.
I can’t speak for everyone, but that’s what happens to me when I try to place myself as either male or female. I could stand up and say “I am a man,” and know, to my bones, that I was lying. Just as I’d be lying if I said “I am a woman.”
It’s not a matter of thinking, “I can’t be a man/woman if I want to do or like these things.” I know that as a woman, I could still have a career, join the military, roughhouse, be athletic, be great at science – all those stereotypically male things. I know that as a man, I could still stay at home, raise kids, bake, knit, show my emotions easily and often – all those stereotypically female things. My gender identity is not about what I want to do, it’s about who I am.
This is not a new idea. Cultures across the globe have acknowledged more than two genders, from the Middle Kingdom of Egypt to the Lakota of North America, from Mayan civilization to the Siberian Chuckhi. References to persons neither male nor female date back to some of humanity’s most ancient written records, such as the Sumerian creation myth, and survive in seminal religious texts such as the Ramayana and the Halakha.
If you want to learn more, the citation list on Wikipedia’s article on “Third gender” has links and references to scholarly articles, books, studies, and excerpts which might help you get an idea of the nature and history of various non-binary identities. Or you can look at more contemporary accounts, such as Neutrois.com, or the discussion on AVEN’s site on “What it feels like to be trans, genderqueer or genderless”. Remember that no single narrative will be able to represent all people, and different nonbinary people may have different preferred terms, explanations, and experiences.