Androgyny is a state consisting of both masculine and feminine gender traits.
In binary gender systems, it might be construed as an "in between" state, much like how bisexuality is construed as being somehow "in between" straight and gay in sexual orientation systems that only take the sex of the object of desire into account.
Non-binary ways of looking at gender make androgyny into more than a simple mixture of masculine and feminine. "Genderfuck", drag, "neutrois", and even physical conditions such as intersex or castration all represent very different approaches to gender, yet under binary systems, they would be lumped under the heading of androgyny.
Sometimes androgyny is entirely outside the bounds of the gender binary of masculinity/feminity. And sometimes it's a fairly straightfoward mix of the two. Whether this mixture is considered outlandish and shocking, or a bland obliteration of gender differences, depends upon both the person presenting their gender in such a manner, and the viewer. Of course many people who consider themselves androgynous are looking for neither reaction. It's just how they are...just as some people are simply men or women.
Overall, androgyny responds to the gender binary system which demands an "either/or" answer with reply of "both" and "neither". Androgyny's relationship to transgender is complicated by the historical existence of "third gender" options in many cultures, which modern Westerners have often seen as a particular culture's way of catergorizing transgender people.