Beef Stroganoff is a dish consisting of thin slices of cooked beef and sliced mushrooms, cooked by sautéing in butter, then mixed into a sour cream sauce.
The beef used needs to be from a good, tender cut, such as tenderloin. It shouldn’t be cooked for too long or it will get tough.
In modern times, Beef Stroganoff is usually served with straw potatoes. This habit started around 1912.
In North America, it is often served instead over egg noodles.
Some North American versions add tomatoes or tomato paste. Other North American versions make it from a can of cream of mushroom soup, with hamburger (sic.)
Instead of beef tenderloin, you can use strip steak or top sirloin.
There are at least two popular theories about how Beef Stroganoff originated.
One is that it was created in 1891 in St. Petersburg, Russia, by Charles Brière, a cook who worked for Count Pavel Alexandrovich Stroganov. Brière reputedly submitted the recipe in that year to “l’Art Culinaire” (presumably the magazine whose full name was “La Revue de l’Art Culinaire”.) This is the version proposed in the 2001 version of the English language Larousse Gastronomique. If this is so, it would seem to be just about Brière’s only claim to fame. His recipe called for shallots (now onions are used.)
The second is that it was created by an unknown cook for Count Grigory Stroganov (1770-1857), because the Count had lost his teeth and couldn’t chew meat.
Beef Stroganoff, though, is probably just a more refined version of similar, pre-existing recipes.
One, for instance, appeared in the 1871 edition of “A gift to young housewives or a help to reduce housekeeping charges” by Elena Molokhovets, in which she based it on a bouillon. Hers didn’t include mushrooms or onions. In her recipe, you make a roux from butter and flour, add some bouillon to make a sauce, and season with some pepper and a teaspoon of mustard, then cook the sauce till thick, and set aside. You rub the sliced beef with ground allspice, sauté the beef in some butter until cooked, add it to the sauce, reheat the sauce, stir in 2 tablespoons of sour cream, and serve. 
The first appearance in English seems to be in 1932, in the book “Good Food” by Ambrose Heath.
It then appeared again in 1939 in the Mystery Chef’s Own Cook Book by John MacPherson (he calls for Worcestershire sauce.)
In 1943, the recipe showed up in the Joy of Cooking by Irma Rombauer, calling for mushrooms.
Beef Stroganoff was very fashionable in the 1970s.
 Recipe 635 in the 1861 edition.
Literature & Lore
“Unlike the French, who name dishes after the chefs who devised them, the Russians have usually attached the names of famous households to their cuisine — the cooks were usually serfs. For example, we have Beef Stroganoff, Veal Orlov, and Bagration Soup. One of the few exceptions is a cutlet of poultry named after Pozharskii, a famous tavern keeper…
The last promient scion of the dynasty, Count Pavel Stroganoff, was a celebrity in turn-of-the-century St. Petersburg, a dignitary at the court of Alexander III, a member of the Imperial Academy of Arts, and a gourmet. It is doubtful that Beef Stroganoff was his or his chef’s invention since the recipe was included in the 1871 edition of the Molokhovets cookbook…which predates his fame as a gourmet. Not a new recipe, by the way, but a refined version of an even older Russian recipe, it had probably been in the family for some years and became well known through Pavel Stroganoff’s love of entertaining.”
— Anne Volokh with Mavis Manus. The Art of Russian Cuisine. New York: Collier Books. 1983. p. 266.