One of the great classic French sauces, demi-glaze (or demi-glace) is a rich brown sauce that is itself a base for many other sauces. Making it is not exactly a walk in the park, for it is a combination of an Espagnole or brown sauce, beef or veal stock, and Madeira or sherry, which is reduced by half.
So the first order of business is the Espagnole sauce. A couple of carrots, celery and onions are chopped into a mirepoix, and sautéed in butter until the onions are golden brown. Add 2 tablespoons tomato paste and continue cooking gently. Make a brown roux over low heat in a heavy pan with 1 cup of butter and 1 cup of flour. When the roux is a hazelnut color, add 6 cups hot stock, and whisk together. Add the vegetables, as well as a bay leaf, a little thyme and some parsley stems. Simmer the mixture for 2 to 3 hours, skimming off the scum that rises to the surface. Strain it through a fine strainer, and press the mirepoix gently to extract their juices. (For step-by-step instructions, follow this Espagnole sauce recipe).
Then, for the demi-glace, add an equal amount of beef or veal stock, bring it to a boil, and simmer until the sauce is reduced by half, again skimming the surface as necessary. Off the heat, add 2 tablespoons Madeira. The sauce can be kept in the refrigerator for a week or frozen for 6 to 8 months.
Having said that, these sauces are seldom made any more, even in restaurants. The more common alternative is Jus Lié, a stock quickly thickened with cornstarch, potato starch, or arrowroot. (Heat 3-1/2 cups of stock to the boil. Mix 5 teaspoons of starch with 1/2 cup of cold stock, and mix it into the simmering stock. Simmer the mixture until thickened and clear.) The quality of your stock is what will make your jus lié work or not. That is also true of the traditional sauces, but it has more to hide behind in the demi-glace.
If you're looking for the middle ground, here is another demi-glace recipe - an alternative that falls somewhere between the traditional sauce and the quick-and-dirty jus lié.