Genderqueer is an umbrella term for people who feel that their assigned gender does not match their actual gender. Definitions in this area are muddy at best, but in general genderqueer parallels the term transgender.
Insofar as there is any difference between the two terms, transgendered has an overtone suggesting that the individual may be specifically trying to make their body fit their perceived gender, especially in the cases of a man who feels that he is female or a woman who feels that she is male. This may be done through cross-dressing, hormonal therapy, voice therapy, or surgery, among others. As such, there may also be a strong tie to sexual orientation.
Genderqueer was in part coined in order to move away from this idea that the differently-gendered are always trying to move from their birth gender to the opposite cisgender, and perhaps also from the idea that gender was so closely tied to sexual identity. However, both terms do technically cover the same subcategories in common usage. Genderqueer identities include, but certainly are not limited to:
In addition to gender identities, humans tend to be obsessed with sexual identities: differences in this area are referred to under the umbrella term non-heterosexual. The terms LGBT/LGBTQ cover the entire spectrum of non-heterosexual and transgender/genderqueer. There are some terms that modify gender roles, such as quirkyalone or metrosexual, that are generally not considered iconoclastic enough to be genderqueer, but still do their part to help gender roles conform to people, rather than vice versa.
Feel free to /msg me with corrections/additions, but I do not plan to try to list every one of the many gender identities in this writeup, nor is this the best place for technical discussions of specific genders.