Some sauces, especially in the rarefied world of haute cuisine, are carefully constructed from a well-established set of ingredients assembled in a time-honored fashion. Others are more casual, representing a broad style rather than a single, canonical result. The brown sauce used in Chinese-American restaurants is an example of the latter: a simple combination of typical flavors that's as versatile in its way as the tomato sauce in Italian-American "red gravy" joints.
Chinese cuisine is equipped with a large repertoire of condiments, sauces and seasonings with rich, concentrated flavors. Two of the most important are soy sauce and oyster sauce, and these form the basis for most versions of brown sauce. In fact, it's helpful to think of brown sauce as a way to dilute these potent flavorings. Once transformed by the addition of extra liquids and a thickener, the thin, dark soy sauce and thicker oyster sauce form a clinging, glistening sauce that lends flavor and savor to a wide variety of dishes.
Your choice of soy sauce has a direct bearing on the final sauce's flavor, so it's worth a moment's extra scrutiny. If your day-to-day brand is in the lighter Japanese style, your sauce will be correspondingly delicate. Bolder mainstream Chinese-style sauce, especially if it's dark soy sauce, has a stronger though less subtle flavor. Artisanal, traditionally brewed soy sauce has a much more complex flavor, which is somewhat at odds with the sauce's utilitarian character but can make a superior brown sauce for special occasions. Mushroom soy and sweeter, thicker hoisin also alter the sauce's character, and can be used to fine-tune it for specific dishes.
The liquid you use to thin the soy and oyster sauces will also play a role in determining the sauce's final character. For simple versions, plain water will work perfectly well. Chicken broth is another common choice, suitable for use in most dishes. Beef broth is relatively rare in Chinese restaurants, but you can certainly use it at home when preparing a beef dish. Add a splash of Chinese shaoxing wine or dry sherry, its Western equivalent, to lend the sauce a bit of extra depth and character. Feel free to add a pinch or two of sugar, when appropriate, or other flavorings such as ground pepper or five-spice powder.
Completing the sauce is as simple as whisking the ingredients together, along with enough cornstarch or arrowroot powder to thicken it, and then applying some heat. You can prepare the sauce in advance if you plan to binge on Chinese cookery over a period of days, by simmering the ingredients in a small saucepan until they thicken. More commonly -- since the sauce comes together in a matter of moments -- you'll whisk the ingredients to form a slurry and then stir them into a near-finished dish in your skillet or wok. The brown sauce will meld with the cooking juices in your pan as it boils and thickens, incorporating their flavors into the finished dish.