What turns broth and drippings into gravy, and milk into white sauce? What makes it tomato soup instead of hot tomato juice? What gives these things that little bit of extra body and thickness that makes all the difference? In a word, starch.
There are several starchy ingredients that can thicken soups, gravies, and even desserts. They’re all handy once you get used to using them, though none of them can work miracles. (And yes, you could always stir in some canned soup or powdered mix instead — but you’d be doing it just for the sake of the thickening agent. Why not just find out what that is and use it straight?)
Flour is probably the all-time favorite thickener, maybe because nearly everyone has it on hand nearly all the time. It’s not the easiest thickener to use, though, or the healthiest. You can’t just add it to your liquid; it’ll clump instead of dissolving. So you’ll need to heat some fat (preferably butter or another animal fat, like lard, though oil will do in a pinch) and carefully blend an equivalent amount of flour into it. When the mixture is smooth, you can add your liquid (for how much, see this article) — which should also be hot, so this process calls for plenty of caution all through.
A big exception to this is blending flour, which I recommend you keep around. It’s processed to dissolve in liquid instead of clumping like regular flour, so you can skip the fat. Still, I’ve found it’s best not to stir it directly into the dish. Instead, mix a little of the hot liquid into the flour until a thick liquid forms, then blend that into the rest of the liquid. You’ll need one and a half to two tablespoons for every cup of liquid.
Cornstarch is a rarer ingredient to have around, but it’s easier to use than flour — just mix it with a little cold water, then add it to the hot liquid. According to my Joy of Cooking (I myself tend to eyeball things instead of measuring), a tablespoon will thicken 1 1/2 to two cups. Cornstarch leaves the sauce translucent and less pasty looking, which can be important. You’ll often see it called for in Chinese recipes.
There are other thickening agents, like arrowroot, which can be harder to find and which I admit I’ve never tried. If you have gluten issues you’ll want to check out something besides flour, obviously. But for most home cooks, flour or cornstarch will thicken the gravy, sauce or soup as much as you need.