To understand this term it's good to know what the individual words mean in the context that they are in.
Yaoi is the Japanese word for male homosexuality. Put simply it means "gay". The word is actually an acronym that comes from the Japanese phrase, "YAma nashi, Ochi nashi, Imi nashi." meaning no climax, no point, no meaning. Yaoi has also come to describe a particular genre of Japanese shows (anime) and comics (manga) that depict male homosexual relationships. These types of shows are becoming increasingly popular among (mostly) female fans of anime and manga. Fans of yaoi often take their favorite male characters, from any anime or manga, and pair them together (regardless of sexual orientation in the original story).
The second word, lemon, is a term used amongst the fanifiction community. Quite frankly it means that there will be a graphic sex scene in any story designated as a lemon. These types of stories would be ratedp either NC-17 or X by the MPAA. Usually authors will inform the reader of any lemons at the beginning of a story. Lemons refer to any type of sexual activity that includes penetration of some sort. Another term that is used alongside lemon is "lime". This is simply a less graphic description of sexual acts. These stories usually do not go further than heavy petting. Lime's are also more commonly referred to as "shonen-ai" meaning boy's love in Japanese.
So if we combine the two words the meaning is: a story that graphically describes the sexual acts of two male participants whose characters usually come from Japanese anime of manga. Typically these types of stories focus completely on the sex scenes and consist of little-to-no actual plot.
A yaoi lemon: "Sasuke quickly shoved his cock into Naruto's ass hitting the fox boy's prostate and making him moan."
A yaoi lime: "Sasuke grabbed Naruto and desperately pressed his lips against the other boy. He could already feel himself growing excited and began to grind against Naruto."