I think that the Urban Dictionary says it best:

1.

Genderqueer is most commonly used to describe a person who feels that his/her gender identity does not fit into the socially constructed "norms" associated with his/her biological sex.
Genderqueer is an identity that falls anywhere between man/boy/male and woman/girl/female on the spectrum of gender identities.
Mary doesn't feel that s/he fits the mold of any one particular gender; therefore, Mary identifies as genderqueer.
2.

Any position in a wide variety of gender identities, spanning the spectrum between male and female. A person who is Genderqueer is not transgendered, though the option is open to transition. Being genderqueer has no bearing on sexual identity or orientation.
Eli might have a woman's body, but ze doesn't feel like a woman or a man. Eli is genderqueer.
3.

"Genderqueer" describes a category of gender that is neither traditionally "female" or "male." Many different, nontraditional gender identities fall under the umbrella term "genderqueer." Genderqueer individuals may identify as having a gender somewhere between female and male or, more commonly, out of the gender binary all together.

Unless told otherwise, polite pronouns for genderqueer individuals are the singular they and ze/hir pronouns. Most genderqueer individuals take offense being referred to as "it."
"Is Dakota a girl or a boy?" "They're genderqueer!"

"So, because she's genderqueer, it means she's a slut and sleeps with everyone, right?" "Haha, no. Genderqueer just means their gender identity. It has nothing to do with their sexual orientation."

"I'm a dude, so I don't really want to take home ec." "Dude, I'm genderqueer! I'll take whatever the hell I want and not worry about it."

4.

The radical notion that a gender identity can be articulated imprecisely enough that it discourages people from trying to police it.
"Gender as in gender. Queer as in to disrupt. Genderqueer as in disrupting the ability to police this gender."

5.

An identification of gender in which people present their gender in a way that does not represent either man or woman, and which allows for open ended interpretations of one's identity. A classification within the transgendered (not to be confused with transsexual) umbrella
n: "that genderqueer is scaring the nuclear family next door!"

a: "I can't pick what gender I want to be today! I guess I'll be genderqueer"