Before going into specific identities, it's important to know what terms like gender identity and nonbinary refer to:
Gender Identity (noun):
One's internal sense of being male, female, neither of these, both, or other gender(s).
Sex Assigned At Birth (noun):
The assignment/classification of people at birth as male, female, or intersex/other, often based on physical anatomy at the same and/or karyotyping.
Gender Expression/Presentation:
Physical manifestation of one's gender identity through clothing, hairstyle, voice, body shape, etc. This does not necessarily indicate gender identity.
Transgender (adjective), aka trans:
Umbrella term for those whose gender identity differs from from their sex assigned at birth.
Cisgender (adjective), aka cis:
Term for someone who exclusively identifies as their sex assigned at birth.
(Neither of the previous two definitions are indicative of gender expression, sexual/romantic orientation, hormonal makeup, physical anatomy, or how one is perceived by others in daily life. Note the lack of "ed" at the end of either.)
Queer (adjective):
General term for gender and sexual/romantic minorities who aren't cisgender and/or heterosexual/heteroromantic. While there is a lot of overlap between queer and trans identities, not all queer people are trans and vice versa. Queer is a mostly reclaimed slur, but one should still be careful when using it.
The Gender Binary (noun): An oppressive system of viewing gender as consisting solely of two, opposite categories ("male" and "female"), in which no other possibilities for gender or anatomy are believed to exist. Many cultures have a traditional Gender Binary, but not all do.
Nonbinary (adj), aka non-binary, non binary, nb, enby: Umbrella tern for all genders other than female/male. Not all nonbinary people identify and transgender, and vice versa. This term can also be used to describe the aesthetic/presentation/expression of a person.
Genderqueer (adjective), aka gq: Umbrella term that's similar to but not the same as nonbinary. It's commonly used by people who don't identify/express their gender within the gender binary. Those who identify as genderqueer may identify as neither male nor female, may see themselves as outside of/in between the binary gender boxes, or may simply feel restricted by gender labels. Many genderqueer people are cisgender and identify with it as an aesthetic. Not everyone who identifies as such also identifies as trans and/or nonbinary.
The "Examples" page has some examples of nonbinary gender identities.