First off, the Wiki links Flan just provided should help.
But to answer the more specific questions--
Most of the terms don't have very solid, set-in-stone definitions, especially when you venture into non-binary (androgyne, genderqueer, etc.) territory, but here's the gist of it...
Genderqueer is an umbrella term for anyone who falls outside the male/female (man/woman) binary, whether they identify as both, neitehr, or something else entirely. Androgyne is along the same lines as genderqueer-- some people only use it to describe people who are "a little bit of both," but more often it's just used synonymously with genderqueer.
Alternatively "androgynous" can describe someone whose cappearance blends masculine and feminine qualities, even if their actual gender identity is one or the other. The key to this is that a person's gender identity is what they feel on the inside, whereas their gender presentation is how they express themselves on the outside. The two often "match", but not always. There are plenty of masculine/androgynous women who are still women, feminine/androgynous men who are still men, and even masculine/feminine androgynes who are still androgynes.
Bigender is kind of a subcategory of androgyne/genderqueer. Usually it's used to describe someone who feels they switch between feeling male and feeling female, though the definition varies a lot from person to person. I've personally never heard trigender before, though, so I can't help you much, there.
As for transman/transwoman-- Basically, a transman is a man who wasn't born physically male (i.e. female-to-male), and transwoman is a woman who wasn't born physically female (i.e. male-to-female). Even if a female-to-male guy doesn't pass (i.e. look convincingly like the gender they identify as), he's still a transman. Likeswise for transwomen. They might be an androgynous-looking transman or transwoman (whether intentionally or unintenionally), but they aren't genderqueer unless they identify as such.
HOWEVER, just to throw a bit of extra mind-boggling into the mix-- a person can be a transman/transwoman AND an androgyne/genderqueer at the same time, if that's how they identify! There are actually quite a few of us around here who identify as such (myself included). To use myself as an example (because obviously I'm most familiar with my own situation), I identify as both FtM and genderqueer. I feel, for lack of a better way to describe it, like a male androgyne stuck in a female body. Had I been born male, I would still consider myself to be transgender on account of being genderqueer. However, I was born female on top of all that, so I guess I get double-whammy queerness? 8D;
