A roux is flour and fat cooked together, usually equal parts. This is done to thicken and flavor sauces, gravies and stews.
There are five stages of a roux: light or blond, peanut butter, medium, dark, and brick. A light roux has little of the nutty flavor that a roux can bring to a dish, and has more thickening power than a dark one. Where as, a brick roux has tons of flavor and very little thickening agents left.
Clarified butter, vegetable oil, and lard are common fats used in a roux. The type of fat that you use depends on the the color of roux that you're going to make. If the roux is going to be light, use butter. The butter will add flavor to your sauce. It will also burn quicker than vegetable oil or lard, so it shouldn't be used for a dark roux. Fats that have a high smoke point are best for dark roux.
When you cook the flour and fat together, you should do it over very low heat and whisk or stir with a wooden spoon continually. Do not leave the stove! A blond roux can take 10-20 minutes and a dark one can take up to an hour to make. There is a thin line between a dark roux and a burnt one. Take your time, and do not burn the roux! If you do burn it, it will be bitter and have black specks. You'll have to throw the burnt roux away and start over.
Once your roux is the color that you're looking for, add aromatics (onion, garlic, peppers, celery, and/or carrots) to the pan cook for a few minutes, then add liquid. This is a great start for any sauce, gravy, gumbo or etuoffee.