One US cup is 236.5882365 millilitres. The long tail of digits is not precision theatre — it is what you get when a unit is defined by a chain of older units rather than by a round number, and it is also why no two recipe sites quite agree on what a cup is.
About the units
The US customary cup is defined as 8 US fluid ounces, and the US fluid ounce is 1/128 of a US gallon. The US gallon in turn is defined as exactly 231 cubic inches, a figure inherited from the English wine gallon of 1707. Follow that chain down and the cup lands on 236.5882365 ml exactly. But there are at least four other cups in working kitchens. The US legal cup, used on nutrition labels in the United States, is defined as exactly 240 ml. The metric cup used in Australia, New Zealand and much of Europe is exactly 250 ml. The imperial cup, still found in older British and Canadian recipes, is 284.13 ml. And the Japanese cup, the gō-derived measure in most Japanese recipe books, is 200 ml. A recipe that says "1 cup" without saying which one has a built-in uncertainty of about twenty per cent.
The exact factor
The derivation runs: 231 in³ per gallon × 16.387064 ml per in³ = 3785.411784 ml per gallon; divide by 128 for the fluid ounce, giving 29.5735295625 ml; multiply by 8 for the cup, giving 236.5882365 ml. Every step is exact, which is why the final figure terminates rather than repeating. In practice almost everyone rounds to 237 ml, and American cookbooks frequently round further to 240 ml — which is the legal cup, not the customary one, so the two conventions quietly converge in print even though they differ by 1.4 per cent.
Where you meet this conversion
This is the conversion that decides whether a cake works. Volume measures are forgiving for liquids and unforgiving for flour, where how you fill the cup — scooped, spooned, or sifted — changes the mass by up to twenty per cent regardless of what the millilitre figure says. That is why serious baking recipes have moved to grams. For liquids the conversion is reliable: a cup of milk, water or stock really is about 237 ml. For anything dry, treat the cup figure as an approximation and weigh if the recipe depends on ratios, as bread and pastry do.