A "Cocktail" is a specific type of mixed drink, and was initially defined as “Spirits of any kind, sugar, water, and bitters”. This would have been the basic recipe that would be used any time somebody asked for a cocktail until the later half of the 1800’s when cocktails might have the sugar replaced by a fortified wine like vermouth, or augmented with a little citrus. Bitters however would still be at the core of almost every cocktail. This would be the case clear up until Prohibition.
Examples of a "cocktail" include:
You can see a list of the cocktail recipes on this site here.