A Lime is a Fan Fic in which characters engage in limited sexual activity, without it being portrayed explicitly. Basically, the fanfic equivalent of soft-core porn. The term is less common than it once was, though by no means extinct.
Can be used as either a noun or an adjective: "It's a Lime story" and "It's a Lime" are functionally equivalent. The boundary between Lime and Lemon is not fixed: a reader unaccustomed to sex scenes may see a fic as a Lemon when a more experienced reader would consider it a Lime. But the rule of thumb seems to be that Lime can generally be expected to keep scenes of explicit intercourse entirely off-screen.
The term was originally specific to the anime fanfiction community, although by the early-middle 2000s it had spread to other fandoms alongside the term lemon. Its origin is obscure: most likely, it was formed by analogy to "Lemon." At one time there was an entire "Citrus Scale" with various fruit code names (Citrus —> Orange —> Lime —> Lemon —> Grapefruit).
For the love of everything decent in this world, No Examples Please.