It's not really a theory, per se. Data shows that for every subsequent son a mother has, his chances of being/turning out gay increase by roughly 33%. It's believed that this has to do with hormonal changes in the womb that become greater when she's carrying a boy, but girls don't seem to change anything. This is why you often see twins with one turning out gay and the other straight, despite presumably being "nurtured" the same way.
Hope that might have answered some of your questions. For what it's worth, most of the gay guys I know are firstborns, myself included. =)
It's still a theory in both senses of the word. A quick search turned up Bearman and Brückner (2008) and Francis (2008) who find extremely weak or no correlation between the two in their data analysis.
I have an older brother and sister, but of the two gay guys I knew from my high school one was a single child and the other had a sister. Human sexuality is far more complex than birth order.