What does the term "86'd" relate to?

What does it mean when someone or something is referred to as being "86'd"?

Without further context, I would take it to mean that that someone or something was rejected, thrown out or discarded. It's a slang expression, encountered primarily in restaurant context. When you eighty-six someone, you refuse to serve them.

Edit: Wiktionary lists a few more meanings, along with this bit about etymology:

Origin uncertain. The [Oxford English Dictionary] suggests possible rhyming slang for nix. Other more elaborate theories include Delmonaco Restaurant in New York City, as item #86 on their menu, their house steak, is supposed to have run out often in the 19th century; another theory is that this term came from the New York speakeasy Chumley’s, which was a hotspot in the 20s. Chumley’s is hidden inside a west village building which has two entrances, a well set back main entrance on Barrow Street and an obscure back-door exit on 86 Bedford Street. When police were sighted approaching the main entrance, the barkeeps yelled ‘86-it' to hide the liquor and signal the patrons to quickly exit the back door.

The Maven's Word of the Day has some additional discussion:

The ultimate origin of eighty-six is unknown. The most widely accepted theory is that eighty-six is rhyming slang for nix[.] One problem with this theory is that rhyming slang has never been very popular in the United States. Another problem is that it doesn't account for the origin of other numeric codes such as [eighty-two, ninety-five, and ninety-nine], which would seem to be related to eighty-six.

Cecil Adams, author of the "Straight Dope" columns, is usually pretty good on etymology and you can read his take on "86" here. His conclusion that it is restaurant code is compelling because there were other code numbers as well, to wit:

Other lunch counter code numbers (I rely here on the Morris Dictionary of Word and Phrase Origins) include 82, I need a glass of water (80 and 81 at times meant the same thing); 99, the manager is on the prowl; 98, ditto for the assistant manager; 33, gimme a cherry-flavored Coke; 55, I crave a root beer; 19, I yearn for a banana split; and 87-1/2, check out the babe over yonder.

Rhyming slang is not a feature of American English and the suggested cognate with "nix" is non-compelling, except that it actually is thinkable with respect to a restaurant number code.

My husband and I both have extensive experience in the restaurant business, and when something is "86'd," it means it is no longer available. As in, someone would say "86 garlic mashed potatoes" — and it would mean we are out of garlic mashed potatoes for the night. I've heard several different stories on how this term came to be associated with running out of something but a lot of them just seemed like industry myths.