Bartending/Cocktails/Martini

The Martini, more so than any other drink, embodies the class and style of the cocktail in the popular imagination.

Ingredients

3 oz. (5.5 cl) Gin
1/2 oz. (1.5 cl) Dry (white/blanc) vermouth (preferably Noilly Prat or Dolin, alternately Martini & Rossi) with ice and water

Either:

Cocktail olive, queen size at minimum. Or olive juice could be added to make it a 'dirty martini'; or,
A twist of lemon peel (including the pith);

Fill a martini glass with ice and water, set aside.
Fill the glass half of a two-piece cocktail shaker to the top with ice, add 1 oz. dry vermouth.
Fill the shaker again with ice, add 2 oz. gin, stir gently.
Discard ice and water in the martini glass.
Strain martini into glass.
Garnish with one olive (two makes it a Franklin).

Adjust the gin/vermouth ratio for a drier martini. The trend in the latter part of the 20th century was to go "ultra-dry" by limiting the amount of vermouth to smaller and smaller quantities. However, this limits the drink to being essentially a cocktail glass of cold gin. The additional herbal and tempering qualities of vermouth are necessary for the proper flavor profile. A 6:1 gin to vermouth ratio is considered "ideal" though your taste may vary.
Replacing the green olive with a pickled cocktail onion turns the Martini into a Gibson Cocktail.
More extreme variants introduce triple sec and/or bitters, but the "classic" recipe stands as merely gin and vermouth.
If you want to be James Bond, the classic Fleming recipe involves an ounce and a half of vodka, an identical portion of gin, and a half-ounce of Lillet Blanc, shaken over ice and served with a twist of lemon.
Stuffed olives go well with a martini. The "Spanish" olive, green stuffed with pimento, is the default. However, those olives stuffed with a garlic clove or bleu cheese are popular as well.
Adding a splash of the olive brine to the drink makes it a "Dirty Martini." Some prefer the additional element of saltiness that this adds.
A Churchill martini involves shaking 6 parts of gin, straining into a cocktail glass, looking at the bottle of vermouth, and serving with an olive garnish.

The Martini is a member of a family of cocktails sometimes referred to as the French-Italian cocktails. Other examples are the Manhattan, Rob Roy, and Harvard Cocktail.

The best gins for a Martini are those with the most complex botanicals -- Bombay Dry, Boodles, Junipero, for example.