There are a bewildering number of opinions as to what constitutes Russian Dressing, so many in fact that one might be tempted to say that this can't be considered an actual, defined item in the kitchen.
But then, without Russian Dressing, there'd be no such thing as a Reuben Sandwich or a Crab Louie Salad, so someone out there must be sure that what they're using is "Russian Dressing."
The most common versions are based on mayonnaise and a tomato sauce. A quick and dirty version of Russian Dressing is mayonnaise and tomato ketchup. A step-up is mayonnaise and a chunky chili sauce, which starts to add texture and interest to the sauce. Better versions seem to go the next step of adding minced pimiento, minced green pepper, minced onion, etc. Some recipes go beyond that even and add chopped boiled egg.
But you can't rely too much on the mayonnaise component: some recipes use salad oil and vinegar in an emulsification (sometimes red wine vinegar) as the base, to which is added the chili sauce, minced onion, chopped egg, etc.
A relatively safe definition of Russian Dressing would seem to be a somewhat creamy, pinkish sauce, based on oil and vinegar -- whether the oil and vinegar come in the form of pre-emulsified mayonnaise, or whether they are emulsified while making the dressing. Egg is very often a component -- whether in the mayonnaise, or whether added in chopped form, or both. There is most likely to be some kind of tomato component to it, giving it both part of its taste, and its colour. Generally, better versions should have somewhat of a sharp taste to them, either from ingredients such as chili sauce, Tabasco, or pimiento.
Bottled versions are made by companies such as Kraft, Seven Seas and Wishbone.
In addition to of course being used as a salad dressing, Russian Dressing is used as an ingredient in many dishes.
Russian Dressing bears no relation to Russian Salad.
Russian Dressing is not and never has been a Russian recipe. It is unknown in Russia; it was invented in America, sometime in the very late 1800s, when many things Russian were fashionable, such as a leather-tanning process that also was referred to as Russian Dressing. 
The recipe for Russian Dressing, though, is entirely American. Disregard anything you hear about the recipe ever having had caviar in it. The recipe was used from the start by ordinary householders.
The first printed version of the recipe we have found, dating from 1900, contains no tomato or chili sauce, though it calls for Tabasco, which would give the dressing a piquant, pinkish hue. This recipe also seems quite thin, as opposed to creamy, though matters get confused when it is served with a mayonnaise sauce poured over it.
"TRIPE SALAD. Buy a pound of prepared tripe (the butcher now sells it ready for use), stew until tender, then cut in dice; add a cup of celery, also in dice; mix these thoroughly with a Russian dressing. For the dressing mix a teaspoonful of salt and pepper each with one tablespoonful of onion juice and four of olive oil; add a drop of Tabasco sauce and two tablespoonfuls of tarragon vinegar. This should be put to one side, and just before sending to the table the part of a lettuce should be pulled into convenient pieces, mixed through the dressed tripe and celery, the whole made into a mould and a mayonnaise sauce spread over all. For a pound of tripe there should be a generous half cup each of Russian and mayonnaise dressing." -- For Table and House Column. Frederick, Maryland: The News. 24 March 1900. Page 5.
[Ed: so the next time someone drones on about authentic Russian Dressing, you can now say the oldest reference you're aware of intends for it to be served with tripe. That should close most such conversations down in a hurry.]
By the time the 1910s roll on, the idea of mayonnaise and chili sauce together for Russian Dressing seems firmly entrenched across the county.
"Dish individually on lettuce leaves, and serve with the Russian dressing, which is a mixture of equal bulk of heavy mayonnaise and chili sauce, well beaten together." -- Wallace, Lily Haxworth. Menus and Recipes for a Week from an Expert in Cookery. Syracuse, New York: The Syracuse Herald. 24 December 1911. Page 68.
"If preferred oyster club sandwiches may be served with Russian dressing instead of Mayonnaise, as the addition of the tomato flavor in the chili sauce is particularly agreeable with oysters, either fried or poached." -- Best of Sandwiches. Brookshire, Texas: The Brookshire Times. 22 January 1915. Page 4.
Russian Dressing may have derived its name from a vague similarity to the version of Russian Sauce (there are two) which is made of mayonnaise, caviar, and the mashed coral from lobsters. The coral (egg mass) is red and gives this version of Russian Sauce a pinkish colour. This is just speculation though; there is no documentation to back it.
 Russian Dressing first appears in print in newspapers as referring to a tanning process done to leather, aka Russia leather. This process tanned skins with willow, birch or oak, then rubbed the flesh side of the skin with birch oil. "The new Spring styles in Women's Gloves are exceptionally stylish... Women's one-clasp, out-seam Cape Gloves, with Russian dressing and the new English thumbs and backs." ( Women's Gloves advertisement. New York Times. 20 February 1902. Page 4.) AND "Gloves in glowing ruddy tans and all the latest colorings; mannish, with one large clasp. Russian dressing, that gives a pleasant odor." ("Now the Fall Gloves" ad in The New York Times. 12 September 1903. Page 4.)
Salad DressingsAemono; Boiled Dressing; Caesar Salad; Green Goddess Dressing; Mayonnaise; Miracle Whip; Ranch Dressing; Russian Dressing; Salad Cream; Salad Dressings; Thousand Island Dressing
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Russian Salad Dressing