The term "mother sauce" refers to any one of five basic sauces, which are used as the base for making secondary sauces. They're called mother sauces because they're like the head of a family. The five mother sauces all use different liquids and thickeners, with three using a roux. Roux is then cooked for different lengths of time to create a light or dark colour.
Béchamel Sauce
The most basic of the mother sauces is the bechamel. No stock, no additional ingredients, just simple store cupboard ingredients. Milk, flour and butter are all you need for a basic béchamel.
The sauce is thickened by the mix of these three ingredients to form a white roux. After that, you can choose to add more spices and flavouring. A traditional bechamel has onion, cloves, and nutmeg and simmered until it is creamy and velvety smooth.
Velouté Sauce
Relatively simple, a veloute is made by thickening white stock with a roux and then simmering it for a while.
For a traditional chicken velouté, you can add chicken stock, but there is also a veal velouté and fish velouté. From there you can adapt the velouté to its own secondary sauce based on what protein you wish to add.
Espagnole Sauce
Slightly more complex than the rest, the Espagnole sauce, also sometimes called Brown Sauce. Espagnole sauce is a basic brown sauce and is one of Auguste Escoffier's five mother sauces of classic French cooking. Escoffier popularized the recipe, and his version is still followed today. Espagnole has a strong taste and is rarely used directly on food.
Hollandaise Sauce
Very unlike a traditional sauce, Hollandaise has a silken texture, made by slowly whisking clarified butter into warm egg yolks. So the liquid here is the clarified butter and the thickening agent is the egg yolks. The emulsified sauce can be used for a number of purposes, over eggs, salads and more.
Classic Tomate Sauce
Everybody knows tomatoes and tomato sauce.Traditionally, the tomate sauce was thickened with roux, and some chefs still prepare it this way. But the tomatoes themselves are enough to thicken the sauce.