"strong, stimulating, cold American drink," first attested 1806; H.L. Mencken lists seven versions of its origin, perhaps the most durable traces it to French coquetier "egg-cup" (15c.; in English cocktay). In New Orleans, c. 1795, Antoine Amédée Peychaud, an apothecary (and inventor of Peychaud bitters) held Masonic social gatherings at his pharmacy, where he mixed brandy toddies with his own bitters and served them in an egg-cup. On this theory, the drink took the name of the cup.
Ayto ("Diner's Dictionary") derives it from cocktail "horse with a docked tail" (one cut short, which makes it stand up somewhat like a cock's comb) because such a method of dressing the tail was given to ordinary horses, the word came to be extended to "horse of mixed pedigree" (not a thoroughbred) by 1800, and this, it is surmised, was extended to the drink on the notion of "adulteration, mixture."
Used from 1920s of any mix of substances (fruit, Molotov). Cocktail party attested by 1907.