Let’s get down to it. "Up" means the martini is shaken or stirred with ice, then strained into a stemmed glass—classically, a V-shaped cocktail glass or a coupe. No ice in the final drink. It’s the opposite of a Martini on the rocks, where the cocktail is poured over ice cubes in a short tumbler.
Imagine your friend who drinks coffee. One friend likes their iced coffee with all the ice, slushy and cold until the last drop. Another friend wants a concentrated cold brew, sipped from a small glass, pure and unadulterated. The "up" martini is that second friend—no dilution, no melting water, just the pure essence of cold booze.
It’s also why that elegant, long-stemmed glass exists. Your warm hands would ruin a martini on the rocks in two minutes. But with an "up" martini, you hold the stem, keeping the liquid icy cold and crisp for the entire experience.