"American cheese" has a legal definition in the United States, which requires it to be pasteurized, processed cheese (that is, cheese to which emulsifiers, salt, colourings, and other ingredients have been added to ensure consistency. Most American cheese is a blend of multiple cheeses (Colby and cheddar most usually), with firm texture and consistent melting characteristics.
In order for cheese to properly be called cheddar, it should be made with whole cows' milk, ripened with lactic acid bacteria, curdled with rennet, and then the curds are cut (to diminish moisture content without losing fat and protein), "cooked" (heated to a temperature to encourage bacterial activity) and then drained. Next, the curds are "cheddared"--a unique process in which the curds are set, cut into loaves, stacked, turned, milled, salted and then aged. These processes will further reduce the moisture content of the curd, resulting in a firm cheese. Salt serves to further reduce the moisture, as well as imparting some flavour, and halting further acidification.
Colby is similar, but it does not undergo cheddarring, which increases the moisture content, making for a softer cheese with less acidity. (Monterey Jack is almost identical, but uncoloured).
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--James