Most dictionaries say the term "eighty-six" comes from employees in the restaurant industry. It means the restaurant has sold out of something. This can happen with a really popular dish, or a special of the day. Specials are foods not on the usual list of foods served.
The head cook, or chef, of a restaurant will tell the head server that the kitchen is out of chicken soup, for example. That server might say to the rest of the employees, "eighty-six the chicken soup."
But, if a customer then orders the chicken soup a server will simply answer, "Sorry, we're out of the soup." It would not be considered respectful to say, "It's eighty-sixed."
Eighty-six has also been used in popular culture to mean dismiss or even kill. If someone has been fired from a job they might say, "My boss eighty-sixed me." Sometimes on television crime dramas you might hear a police officer or a suspect say someone was "eighty-sixed," meaning killed.
The Random House Historical Dictionary of Slang provides a possible explanation for the use of eighty-six in restaurants. The notation comes from a nineteen twenties play called "Burlesque." A waiter in the play tells a customer, "If you need any Scotch or gin, sir -- … My number is Eighty-Six." In other words, he is saying, if you run out, call eighty-six.
The same dictionary also says another former definition of eighty-six was undesirable customer. It notes a nineteen forty-six book about the famous American actor Lionel Barrymore. Gene Fowler wrote that Barrymore was considered an "eighty-six" at one drinking establishment, or bar. He was not to be served because he visited the bar often and did not behave well.