Roux has been thickening savory dishes for centuries. Its first incarnation was in France and made with butter and flour. The invention of the technique in the 17th century at the court of the Sun King was a true revolution in French, and thereby European, cuisine, with four out of the five Mother Sauces of classical French cooking based on a roux. This mixture is only heated for enough time to cook the flour, and is the base of many sauces (including white or béchamel sauce) as well as soups and stews.
There are three different kinds of roux. White and blond roux are the most common, used to thicken sauces, soups, and chowders. Brown and dark brown roux have more flavor, but less thickening power than white or blond roux. Dark roux are primarily used in Cajun and Creole dishes, most notably gumbo and jambalaya.
We heat the fat we are using in order to release an important component of the flour, starch. The starch granules thicken by absorbing water in the food, we are trying to thicken, and swelling to many times their original size.
The ratio of flour to butter in a roux is 1:1. For example, if you use 4 tablespoons of butter, you would use 4 tablespoons (or ¼ cup) of flour.
White roux cooks for the shortest amount of time (2–5 minutes) and functions primarily as a thickener in dishes like macaroni and cheese and clam chowder. On the other end of the spectrum, dark brown roux cooks for 30–45 minutes and acts as an integral seasoning in dishes like gumbo and étoufée.