Bechamel sauce is one of four mother or base sauces in French cooking. It uses milk as its base and is thickened with a roux of flour and butter in equal parts, the amount depending upon the thickness desired. A chart is listed below. From this base comes many other sauces including mornay used in mac and cheese and veloute, which is part bechamel and part stock as well as others. In Italian it is balsamella. It is most often referred to as White Sauce in American cooking due to its color.
Chart of Thickness: Per 1 cup of milk, whole or 2%.
Thin Sauce – 1 tablespoon butter + 1 tablespoon flour
Medium – 2 tablespoons butter + 2 tablespoons flour
Thick Sauce – 3 tablespoons butter + 3 tablespoons flour
Method of Making
Milk, butter, flour, salt and pepper are pictured below. I differ with some who feel the milk has to be hot to avoid lumps in the sauce. I have always used cold milk and found the lumps occur if the sauce is not constantly whisked to keep the thickening agent suspended in the milk. Otherwise, it sinks to the bottom of the pan, thickening the bottom of the sauce first, from which it cannot recover to its velvety smooth consistency.
Melt the butter in a saucepan.
Add the flour and whisk to combine.
Whisk for a minute or two, depending upon the amount to combine and “cook” the flour so no raw taste remains. Do not let the sauce brown too much. This is difficult with a small amount of butter and flour as they heat up quickly. If it browns somewhat it is not a catastrophe, just add the milk immediately. 
Continue to whisk constantly to prevent lumping.
Bring to a boil, reduce heat but maintain a boil and cook for one to two minutes to complete the thickening.
Use as directed immediately or, if holding for more than a few minutes, cover directly with film to prevent a crust from forming on top which will ruin the smooth consistency when stirred again.
