A béchamel sauce is one of the four “mother” sauces in the traditional French cooking repertoire. The others - although we're sure you know them already - are sauce velouté (a white sauce made with stock), sauce espagnole (a brown sauce made with stock), and sauce allemande (a velouté that has been thickened with egg yolks). The béchamel sauce is simply a milk-based white sauce. And the good news is, it is incredibly easy to make.
Most people we know take two tablespoons of butter and two tablespoons of flour and cook them gently in a pan for a couple of minutes, stirring occasionally. By cooking them you do away with the taste of raw flour, which some people find objectionable. The cooked butter and flour is called a roux, which means “reddish” or “reddish-brown” in French, and, indeed, you can cook the butter and flour through various shades of brown until it turns mahogany, in which case, it is only good for Cajun cooking. For a béchamel sauce, you don't want to do that. You want to keep the butter and flour mixture white.
In a microwave oven on in a separate pan on the stove, heat a cup of milk. Gradually add the milk to the roux, stirring constantly with a whisk. Cook it gently for a few more minutes and it will thicken nicely. Season with salt and pepper (if you use white pepper, you won't have little black specks in your white sauce).
If you want a thin béchamel sauce, use only one tablespoon each of butter and flour. If you want a thick sauce, use three tablespoons of each. The only danger you confront in this recipe is that you may burn the roux if you neglect it. And if you burn the roux, throw it out and start over.
Now, you can be much fancier with your preparation, and the classic French chefs would like you to be because it produces more flavor. You can add a small onion, a bay leaf, and a clove to the milk and simmer it for 15 minutes before straining it and adding it to the roux, or you can add them to the sauce once it has been mixed and simmer that very gently for 15 to 30 minutes before you strain out the onion, bay leaf and clove.
You can also throw a peeled carrot, a celery stalk, and a bouquet garni into the mix, which, of course, you also strain or fish out before you use the sauce. It is also quite traditional to use a pinch of nutmeg in the sauce, although you must use a light hand, because you don't want that flavor to dominate the sauce.