Yaoi means, literally, "yama-nachi ochi-nashi imi-nashi" which I'm lead to believe means "no climax, no resolution, no meaning." The way it was explained to me is this; it's a way to define a gay (homosexual) story that has explicit content in it. That much I derived from the translation. However, many use it to simply state that, "hey, there's gay stuff in this story, so if you don't like it, go."
Shounen-ai means "boy-boy love." This is a more general term that I would think you could use to describe just a romantic story between two guys without any sexual content in it. This is, of course, my opinion, and either of these two would work just fine in describing a gay story. Most use yaoi, since it's shorter and more well-known, but shounen-ai works as well.