One stick of butter is exactly half a US cup. It is also 8 tablespoons, 4 ounces by weight, and 113.4 grams — a set of coincidences that is not coincidence at all, since the stick was sized to make all four of those numbers convenient.
About the units
The stick is an American packaging convention rather than a unit of measurement: butter in the United States is sold in one-pound boxes divided into four wrapped sticks, each a quarter pound. The wrapper is printed with tablespoon markings so the cook can cut directly on the line. Nothing equivalent exists elsewhere. European butter is sold in blocks of 250 g or 500 g with no markings, and a British recipe specifies butter by weight in grams or ounces. The stick's dimensions vary regionally within the United States too: the long thin Elgin stick is standard east of the Rocky Mountains, and a shorter, fatter stick is standard west of them. Both contain the same quarter pound.
The exact factor
The relationship to the cup is exactly one half, because a US cup of butter is defined as half a pound for this purpose. From there: 1 stick = 8 tablespoons = 24 teaspoons = 4 oz = 113.398 g = 118.294 ml. The last two are different numbers because butter is denser than water — 113.4 grams of butter occupies about 118.3 millilitres. Confusing the mass and the volume is the standard error when converting an American recipe, and it produces a four per cent shortfall.
Where you meet this conversion
Any American baking recipe. "One stick, softened" and "half a stick, melted" are standard phrases, and a European cook reading them needs 113 g and 57 g respectively. The tablespoon markings on the wrapper make partial quantities easy in an American kitchen and impossible elsewhere, which is why converted recipes usually give grams. For the reverse conversion, a 250 g European block is 2.2 sticks, so recipes rarely scale neatly between the two packaging systems.