A demi-glaze, or demi-glace, to use the original French, is a combination of 1 part beef stock with 1 part brown sauce (with a bit of Madeira or sherry thrown in), which is cooked until it is reduced by half. The result is very rich with concentrated flavor, and is used as a base for many other sauces.
The brown sauce, or sauce Espagnole, is made of a rich meat stock, a browned vegetable mirepoix (diced carrots, onions, celery, and herbs), a brown roux, herbs, and sometimes tomato paste.
Making demi-glaze is more work than most home cooks are comfortable with, yet a surprising number of recipes still call for it. It is available commercially by mail order and in some gourmet shops, although the price is high.