A Lemon is a Fanfic with explicit sexual content. This can range from plot-what-plot boinkfests with no justification or explanations to exquisitely plotted and crafted stories that just happen to follow their participants into the bedroom (and through the subsequent action therein) on a regular basis. Once a common term, it's now less frequently seen, though by no means extinct.
Can be used as a noun and an adjective interchangeably: "It's a lemon story" and "it's a lemon" are functionally equivalent. Surprisingly, the term is not automatically synonymous with "bad" (although that, as the saying goes, is often the way to bet).
The term was originally specific to the Anime fanfiction community, although by the early-middle 2000s it had spread to other fandoms as an alternative to just calling something porn or smut. The origin of the term is obscure, but the most common etymology traces it back to the "classic" Hentai anime series Cream Lemon, by way of an early mailing list for publishing erotic anime fanfic, called "The Lemon List" in its honor. The term derives from the idea that like a lemon, sexuality looks beautiful and alluring, but when you experience it more directly, it's bitter and sour.
The trope has nothing to do with the Lemony Narrator, although combining the two would be quite interesting, especially if it were in a Fanfic featuring 30 Rock's Liz Lemon. Possibly shipped with Cure Lemonade.
No examples, please; we'd be here all day. But feel free to list this as a trope on appropriate Works pages.