When we think about gender, we're often expected to think about it in the context of the socially constructed "gender binary" - the common idea that there are only two genders, woman & man - corresponding to the terms female & male, feminine & masculine, respectively - and the consequent assumption that all people fall into one or other of these categories, and that these assigned gender identities remain fixed throughout the course of our lives.
There is plenty of evidence these days that even biologically, in many species including our own, sex is way more of a wide field of possibilities than the simple female-male we're usually led to assume - newborn babies show some form of intersex characteristics as commonly as they do Type 1 Diabetes (so it follows that some non-binary identified people are also intersex). And gender (our sense of identity within that field of possibilities) is just as broad and complex a field.
Non-binary is an umbrella term used to describe the identities of those of us for whom this "gender binary" doesn't fit: our sense of our gender may fall outside the strict categories of woman/man entirely, or may incorporate only parts of one or both, or be more than either one of those on its own - or we simply may not experience our identity in terms of gender at all.
Non-binary people mostly (but not always) consider their identities to fall under the umbrella of trans (a term applied to individuals and groups whose behaviour or gender identity varies from or crosses conventional gender norms) - but they tend to vary from the traditional notion of trans historically perpetuated by the medical establishment and media, who typically describe trans individuals as simply "female-to-male or male-to-female" - which our binary-conditioned society has been only too eager to latch onto as "familiar".
There are many labels that we might use to reflect our non-binary identities, including: transgender, genderqueer, genderfluid, gender-questioning, gender-neutral, intergender, agender, non-gender, bigender, polygender, pangender, androgyne, epicene, neutrois, third gender, two-spirited, non-binary femme, non-binary butch, transfeminine, transmasculine, demigirl, demiboy, grrl, boi, transsexual...
And some non-binary identified people do include "woman" and/or "man" in their sense of gender identity, though typically in a non-traditional, open-ended and non-exclusive sense.
Many people find these kinds of labels very helpful in making sense of themselves and their identities (or in finding others with similar experiences!) - but they're not essential, we don't need to know what to "call ourselves" to know who we are.