A Lemon is a Fan Fic with explicit sexual content. This can range from plot-what-plot screwfests with no justification, 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. Those fond of Lemons refer to it as 'Lemony Goodness'. Surprisingly, the term is not automatically synonymous with "bad", or even "vulgar". It's also subjective. Some people believe that it has only to be penetrative to be a Lemon, some believe that it has to be explicitly penetrative, while others believe that if it goes into enough arousing detail, even reaching second base can be a Lemon. Somebody venturing into sex scenes for the first time may see a fic as a lemon, while a more experienced reader may consider it a Lime.
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.
Some fandoms have their own idiosyncratic terms with roughly the same meaning. For example, My Little Pony fans know Lemons as "clop" or "clopfics" in reference to the brony fandom's favorite Unusual Euphemism for A Date with Rosie Palms.
The trope has nothing to do with the Lemony Narrator, although combining the two would be quite interesting, especially if it were in a Fan Fic featuring 30 Rock's Liz Lemon. Possibly shipped with Cure Lemonade or Lemon Breeland.
Not to be confused with those tasty, tasty fruits, the slang term for a lousy car, the Russian slang term for a million dollars (or a hand grenade), OR pictures of old men having a party (okay, perhaps that last example is related). Has nothing to do with burning your house down with the aforementioned yellow fruits. Nor does it refer to the U.S Supreme Court test on what is excessive government involvement in religion or a famous collection of sites dealing with Commodore 64 games.
Compare Lime, Porn Without Plot. See also Slash Fic, Rule 34. Frequently found in Kink Memes. Music versions of this may fall into the category of Intercourse with You, although not all IWY songs are, strictly speaking, lemons. See IKEA Erotica and Common Hollywood Sex Traits for some of the more common ways these stories can go wrong.
No examples, please. This just defines the term.