Roux (pronounced "roo") is a mixture of equal parts (by weight) fat and flour that is used for thickening sauces and soups. Clarified butter is the type of fat most frequently used.
[Also see: Thickening a Sauce with Roux]
Note that equal parts by weight does not necessarily mean equal parts by volume. Because butter is denser than flour, you would actually need a 2 to 1 ratio of flour to butter by volume. In other words, two tablespoons of flour is equal to one tablespoon of butter by weight.
Roux is made by stirring the flour into the melted butter and then cooking it for a few minutes so that the floury taste cooks off. Then, the roux can be used or cooked further to develop more color and a toasty flavor.
Roux can also be made with other fats such as lard, bacon fat or simple vegetable oil. Likewise, any starchy flour, such as rice flour or potato flour, can be used for making a roux.
The roux can be incorporated by whisking a liquid like hot milk or stock into the roux. Or the roux can be added to a soup and stirred in to thicken it.
Three of the five mother sauces of classical cuisine are thickened with roux: the velouté, the béchamel and the espagnole.
See also: How to Make a Roux