Béchamel is a standard white sauce and one of the five mother sauces of classical cuisine. Simply butter, flour, milk and a bit of salt and pepper, it's elegant but not always easy to get right. It's often included in lasagna and pastitsio as is, but the French usually modify it in some way. Adding cheese produces Mornay Sauce, adding crawfish, shrimp or crab turns it into a Nantua sauce, Soubise features onions sweated in butter, and by grating, in some fresh nutmeg and by substituting cream for the milk it shifts into a cream sauce.
The Trick to White Sauces
The trick to these sauces is that the flour and butter (or any other fat) are whisked together over low heat for about five minutes (don't leave its side). The flour/butter mixture is known as a roux. This step coats the starch granules in the oils so they don't clump together and also cooks the flour so it no longer tastes raw. The flour and butter should be equal weights, but equal quantities also work up to about 1/4 cup of each. The amount of milk determines how thick the sauce is - figure a cup of milk for every 2 tablespoons of butter and flour. So two cups of Bechamel would be 4 tablespoons butter, 4 tablespoons flour, and 2 cups milk. Gradually bring the sauce to a boil over medium-high heat, stirring constantly to prevent burning, for maximum thickening. As soon as it boils, pull it off the heat. (Note: we usually heat the milk - or stock - in the microwave before adding it to speed up the thickening process.)
Let Your Imagination Run Free
At this point, you can add other flavorings such as salt, ground white pepper, perhaps some dried mustard, some finely chopped crawfish or lobster, sweated onions. Let your imagination run free.
You don't have to stick with either butter for the fat or milk for the liquid. Fry up some crumbled breakfast sausage, sprinkle it with flour when it's nearly done, and add milk 5 minutes later to make sawmill gravy for biscuits. Or start with butter and flour, but replace the milk with shrimp stock for a particularly rich Nantua sauce. And of course, you can make a wonderful Mornay sauce by just adding some gruyere, cheddar, Manchego or other cheese. (If you're adding cheese, use a bit more liquid to produce a thinner base sauce, the cheese will thicken it up.)