French chefs Antoine Carême and Auguste Escoffier codified mother sauces in cuisine as roughly five sorts. By “mother” sauce is meant that countless other sauces are derived from these.

Béchamel (besciamella)...

the classic white sauce, was named after its inventor, a steward of Louis XIVth. Called the “king of all sauces,” it is known as a cream sauce and is used everywhere in extended form from sawmill gravy over biscuits to alfredo over pasta.

The basis of this sauce is made by stirring milk into a butter-and-flour roux. A basic a thin sauce consists of one tablespoon each of butter and flour per cup of milk, increasing the amount of butter and flour per cup of milk determines, of course, the thickness. The more roux (brown) the roux becomes, the weaker the thickening effect.

This sauce (or the velouté) is good place to start for chicken gravy.

Velouté...

is a stock-based white sauce made from from chicken, veal or fish stock and a roux. Enrichments such as eggs and cream are sometimes used to create a sauce allemande or sauce blonde, e.g.::

1½ cups velouté

2 egg yolks

½ cup heavy cream

salt, pepper

lemon juice

Espagnole...

is a brown sauce made of a rich beef or veal stock, a mirepoix of browned vegetables, a browned roux, herbs and optionally tomato paste. This is a very complicated sauce that can take days to manufacture if you start from scratch. Most, including restaurants (and even high-end ones) start from an intermediate step.

This sauce is good for starting a beef gravy.

Hollandaise/Mayonnaise...

are both made with an emulsion of egg yolks and fat.

Hollandaise is made with butter, egg yolks and lemon juice, usually in a bain marie to prevent cooking the eggs, and served warm over vegetables and fish. Eggs Benedict makes use of this sauce.

Mayonnaise is a thick, creamy dressing that is an emulsion of vegetable oil, egg yolks, lemon juice or vinegar, challenging to succeed. It is the basis for many other dressings including Thousand Island, Tartar Sauce, Aïli (garlic) and “Fry sauce.”

Tomato...

sauces are legion because tomatoes themselves have rich flavor, low liquid, especially when cored, and their flesh cooks down quickly. Tomato sauces may contain a meat content (bolognese). They are also heavily used in Indian cuisine (and British-Indian like chicken tikka masala).

Other

In addition, there other liquids and syrups used culinarily such as vinaigrettes, cranberry, beurre blanc. Different people categorize sauces differently.

A great reference site for sauces is found at http://lynnescountrykitchen.net/sauc/mothersauces.html.