What does it mean to not identify with either gender?

I am genuinely confused by the concept of not being able to identify with either gender. I understand if someone is unable to identify with the stereotype of either gender; however, both genders truly encompass such a wide range of possible “selves” that I find it difficult to understand how someone would be unable to find a place in a gender that was coherent with him or herself.

I understand most other forms of gender and sexuality. I comprehend most types of sexuality for both men and women, but that really has nothing to do with gender. I understand men who were born in women’s bodies and vice versa. In all these cases of gender and sexuality, people still feel as if they identify with a particular sex, regardless of whom they’re attracted to and what their bodies looks like.

But what does it feel like to not identify with either gender? Is it that they can’t or that they don’t want to because they disagree with the societal pressure to assume a gender role? Do they feel as if they identify half as men and half as women, or do they feel as if they simply identify as “person,” and not as part or either gender? Are all people either “man” or “woman,” or are some people truly neither?

Also, while we’re on the topic of gender, if you have the time, could you define what the following words mean to you (please no links to definitions; I can find them myself):
Intersex, eunuch, bigender, agender, androgyne, genderqueer, third, fourth, fluid, androgynous, genderfuck, cisgender.