Yaoi is a Japanese word, which is actually something of an acronym. It comes from "YAma nashi, Ochi nashi, Imi nashi", which means "no climax, no point, no meaning".
In speech and writing, it's a slang term used to refer to stories that focus on homosexual relationships, usually between men. Yaoi tends to involve graphic sex, often with little or no plot. The female equivalent is yuri.
A related genre is shonen-ai, or "boys' love", which tends to involve less sex and more cuddling (and plot).