The cocktail made with gin, vermouth, and an olive or lemon twist is a Martini.
A classic Martini combines gin with dry vermouth, is chilled with ice, and is strained into a stemmed cocktail glass or served over ice. Its traditional garnish is an olive or a strip of lemon peel called a twist. The exact proportions vary widely, which is why bartenders may ask whether a guest wants the drink dry, wet, stirred, shaken, or dirty.
A dry Martini contains more gin and less vermouth; a wet Martini contains more vermouth. A dirty Martini adds olive brine, while a Gibson uses a cocktail onion instead of the usual olive or citrus garnish. These are variations on the same basic family rather than entirely different drinks.
The vodka Martini became especially famous in popular culture, but it substitutes vodka for the traditional gin base. That distinction explains why “Martini” alone normally points to the gin-and-vermouth cocktail in a standard quiz. The Manhattan is a related vermouth cocktail made with whiskey, while an Old Fashioned contains whiskey, sugar, water, and bitters.