One US cup is exactly 16 tablespoons. Both units are defined through the same gallon chain, so the sixteen is exact, and it is the relationship that makes partial cup measures possible without a dedicated measuring cup.
About the units
The US cup is 8 fluid ounces and the US tablespoon is half a fluid ounce, so sixteen tablespoons make a cup by definition. In millilitres that is 236.588 and 14.787 respectively. The number sixteen is convenient because it is a power of two: halves, quarters and eighths of a cup all land on whole tablespoons. It fails only for thirds, which is why a third of a cup is 5⅓ tablespoons and why American measuring sets include a separate ⅓-cup scoop rather than expecting cooks to count spoons.
The exact factor
The factor is exactly 16. The useful sub-values are worth memorising because they come up constantly in American recipes: half a cup is 8 tablespoons, a quarter cup is 4, an eighth is 2, and a sixteenth is 1. A third of a cup is 5 tablespoons plus 1 teaspoon, and two thirds is 10 tablespoons plus 2 teaspoons. Those last two are printed inside the lids of many American measuring sets precisely because nobody remembers them.
Where you meet this conversion
Scaling recipes up and down is the main case, along with measuring quantities that fall between the marked cup sizes. Butter is the other frequent user, since an American stick is 8 tablespoons — exactly half a cup — and the wrapper is marked so it can be cut to the tablespoon. When converting an American recipe to metric, going through tablespoons rather than directly from cups often gives a cleaner result for small quantities, since 15 ml is a friendlier working number than 236.6 ml divided by an awkward fraction.