So there's two very popular books that define "Six basic drinks":
Cocktail Codex: Fundamentals, Formulas, Evolutions
We're going to label the Wikipedia excepts with the recipe defined by each book.
The six drinks they define are:
Martini
Stir with ice, strain into a cocktail glass, twist lemon peel over the top and serve garnished with an olive, preferably one stuffed with any kind of nut. Embury also states that sherry is a nice substitute for vermouth.
Daiquiri
Shake with lots of finely crushed ice and strain well into a chilled cocktail glass.
Sidecar
Shake vigorously with ice and strain into a cocktail glass. Garnish with a twist of lemon, if desired.
Old Fashioned
In an old-fashioned glass, add bitters to simple syrup and stir. Add about 1 ounce of whiskey and stir again. Add two cubes of cracked, but not crushed, ice and top off with the rest of the whiskey. Twist lemon peel over the top and serve garnished with the lemon peel and a maraschino cherry.
Manhattan
Stir with ice, strain into a cocktail glass and serve garnished with a Maraschino cherry.
Jack Rose
Shake vigorously with ice and strain into a cocktail glass. Garnish with a twist of lemon, if desired.
The 12 Bottle Bar: A Dozen Bottles. Hundreds of Cocktails. A New Way to Drink.