A martini is a human alcoholic beverage traditionally made from a mixture of gin and dry vermouth, with a variant "vodka martini" which substitutes vodka for gin.
Preparation of the martini generally consists of a small amount of vermouth poured over ice to slightly melt and flavor the ice. The desired amount of gin or vodka is added and the mixture is shaken or stirred according to taste and then served without the ice.
L.Q. Clemonds ordered a martini from the replicator of the Enterprise shortly after being awoken from a cryonic satellite in 2364. (TNG: "The Neutral Zone")
In the Julian Bashir, Secret Agent holoprogram, Mona Luvsitt is very adept at making martinis. (DS9: "Our Man Bashir")
Vic's Las Vegas Lounge serves martinis. When Jake Sisko and Kesha joined Nog at Vic's, she ordered a martini. Julian Bashir ordered a vodka martini, stirred not shaken. (DS9: "It's Only a Paper Moon", "Badda-Bing, Badda-Bang")
Also see: Klingon martini