On the social side, it's how we perceive ourselves and how we choose to present ourselves.
On the biological side of the question, in how we perceive ourselves, most of this is in our heads, quite literally, but there are no easy or cheap tests for it, short of a post-mortem on the brain.
What is known about most MTFs is that they have brain structure differences from other males. The differences in brain structure almost always closely resemble the structures in females.
Back in the dark ages of gender identity diagnosis, a doctor at Johns Hopkins (whose name escapes me at the moment) stated that we are born as blank slates and we learn our gender roles so all that matters is how you raise a child. This was put to the test in the infamous "John/Joan" case, where a male with a badly botched circumcision was surgically altered to be a girl and raised as a girl. This, of course failed, and by the age of 14 the child was prepared to commit suicide. Attempts were made to help the child and later as an adult, but as I recall the individual in question ultimately took his own life in his 30s.
Dr. Milton Diamond never believed this theory so he eventually followed up on the "John/Joan" case and discovered that it did not work out the way the theory said it should. And, it turns out, it didn't work out in many other cases as well. Further, Dr. Diamond has stated (in interviews) that scientists have subsequently forced transsexual behavior on animal species by raising the testosterone levels on female fetuses in the womb (and vice versa on male fetuses). In every case, despite being biologically of one sex, these animals would adopt the behaviors of the other sex.
This has led to extensive studies that show that transgender people, going both ways, MTF and FTM, have significant brain structure differences from others of the same biological sex. Since our sense of self is within the brain, we quite literally have female brains in male bodies or male brains in female bodies.
Some of the recent hormonal research is summarized
here, in this presentation to the AMA in 2011. It's a long viewing, over an hour, but I highly recommend it.
Here is one image of just one brain region that is different in MTFs.
The image above is the central nucleus of the bed nucleus of the stria terminalis (BSTc), in the thalamus. Note that heterosexual and homosexual males have nearly identical brain structures in that region. Note that biological females have very different structures in that region from those of the males. And finally note that MTF transsexuals have brain structures that are very close to the female's and nothing at all like the biological male's neurological structures.
So no, you are not just "gay".
And yes, being transsexual is a real medical condition for which there is treatment that works for those who suffer most. That treatment is HRT and transitioning to live in the gender with which you identify.
Reference from my blog:
Biological differences in MTFs and cisgender males
