A) Roux: heat some oil in your fairly thick-bottomed pan, on medium. Add flour slowly, stirring continuously, until you have a paste which is no longer shiny with oil on the surface.
If you want pale roux, you may start to add the other ingredients of your recipe as soon as your paste is smooth.
If you want golden/"peanut butter" roux, keep toasting the flour until the paste mixture is the colour of light toast.
If you want brown/dark roux, keep toasting the flour until the paste mixture is the colour of dark toast, being very careful to keep the heat low and the paste always on the move with your spoon so as not to burn the flour. It takes a while, but it's worth it: burnt roux is nasty.
A pale roux and a dark roux will not have the same thickening power: pale roux will thicken much more strongly, but not have as complex a flavor.
Once your liquids have been added, slowly bring the mixture to at least a one-minute boil to cook the flour completely and achieve your final thickness.
1 lb roux (8 oz flour to 8 oz oil or other fat) will thicken about a gallon of water or other thin liquid. If you think you may have ended up with too much roux, move some or all of it into a bowl and add it in gradually as you cook the rest of your dish until you are satisfied with the thickness, remembering that roux doesn't thicken until it boils.
B) White sauce:
Melt butter in a thick-bottomed pan over very low heat, add flour slowly, a teaspoonful at a time, stirring continuously, until you have a fairly stiff paste with no fat shining on the surface. As soon as you have a smooth, lump-free paste, add
i) cream (or half-and-half, or milk). Bring to a very low, brief, careful boil, just enough to cook the flour without damaging the texture of the cream or milk. You now have a Bechamel sauce, which you may flavour as you like or as the recipe dictates.
ii) chicken or fish (or vegetable) stock. Bring to low boil. You now have a Veloute, which you may flavour as you like or as the recipe dictates.
C) Thickening sauces or other liquids which contain low or no fat: Make a thick paste of soft butter and flour, introduce it slowly to the liquid and bring your dish to a boil to cook the flour.
(If your mother or grandmother or some other older relative or friend taught you how to make gravy, keep right on doing it that way, for that is The Best Gravy. If not, you may as well make it the way my grandmother did it as any other way: it's fairly simple, it makes good gravy, it almost never ends up lumpy, and the method has been tested and refined over at least 100 years of Thanksgivings, Christmases, and Sundays.)
While your perfectly roasted or pan-fried meat is resting/keeping warm in the oven, add a small amount of wine, beer, sherry, port, stock, broth, or water to the pan and bring it to a strong boil, scraping up as much of the bits of dripping and meat and so forth from the bottom of the pan as you can and mixing them into the fat (this is called deglazing, if you want to be impressive. Gran called it 'getting all the good out').
Let that liquid mostly evaporate, then turn the heat OFF and start adding flour slowly - I use a teaspoon except at Christmas, when I use a 1/4 cup measure to deal with ALL THE TURKEY FAT. You turn the heat off to give yourself time to work; you don't want the flour to cook until AFTER you add your liquid or you will get the dreaded lumps.
If you think you have too much fat, or you KNOW you have too much paste, both can be stored for later: the paste can go in the freezer in anything that seals. A method for clarifying and storing fat for later use is here.
When your paste is smooth and no longer shiny, add some meat stock or vegetable stock or the water you used to cook tonight's vegetables, or even last night's, which you poured into something and kept when the vegetables were done, or some boiling water from a kettle or a bit of all of the above, gradually, stirring as you go. Turn heat to medium and continue stirring until mixture boils, thickens, and smooths. Add more liquid if desired to adjust thickness. Add herbs and spices to taste, and a glug of wine if you want to.
E) Alternate flours: most grain flours (whole wheat, rice, spelt), and legume flours (chick pea, pea.) Nut "flours" are not absorbent in the same way and do not work. Non-wheat flours will require different proportions: add your flour slowly and patiently and you ought to be fine.
A) Potato water, from boiling potatoes. Good added to soups, stocks, gravies, stews ... the more you boil it down the more thickening it will give, but it will in no case give very much. Adds a subtle but definite flavour, especially if you boiled them with the skins on, in which case the water tastes a bit earthy. I like this, especially in red meat gravies; you may not.
B) Mashed potatoes: mash them very smooth, then slowly add stock or milk or the liquid from your soup or stew and continue to mash/stir until you have a thick liquid. Stir it slowly into your dish. Adds a decided potato flavour.
C) Potato Flakes: Make sure you have proper flakes, not "instant mashed potatoes", which have "butter flavouring" and salt and milk powder and all sorts of things that will make your dish taste and behave oddly added to them. Sprinkle handsful into your soup or stew. Comparatively, it takes quite a lot to make a difference, so if you want a very thick final result it is more economical to use mashed potatoes, as above. But this method is very quick and easy, and even the flakes are not terribly expensive. Adds nearly as much potato flavour as freshly-mashed potato.
A) Corn Starch, Potato Starch, Tapioca Starch, Arrowroot/Kudzu Root:
Thickens without flavouring and without diluting flavour.
In a small cup or bowl make a slurry of powdered starch and some cold liquid, stirring rapidly with a fork or whisk until your mixture is smooth and thickish.
Add the slurry gradually to your dish, about a spoonful at a time, stirring continuously until desired thickness is reached. As with flour, you won't see any thickening until your dish boils, so start small and don't overdo it. You can always add more and bring it back to the boil, if you need to.
Note: Don't try to harvest your own kudzu unless i) it's on land you control ii) it's free of herbicides, fungicides, and not right by a road where it's been picking up vehicle exhaust and iii) you're willing to research how to process the root, because I do not know how and so cannot tell you. Roots are NOT safer for having been under the ground; pollutants concentrate in them.
B) File/Sassafrass Root: similar to other powdered starches, but with more flavour and much less thickening power. Because it is less thickening and more soluble, smaller amounts (1-4 Tablespoons) can be sprinkled directly into dishes without causing lumps; for larger amounts, make a slurry as above.
Potatoes, Rice, Barley, Kasha, Quinoa, etc: sliced, whole, mashed, or pureed, starchy vegetables and grains thicken the broth of soups and stews as well as adding flavour, nutrients, and bulk.
Simple, easy, often overlooked. Simmer your sauce, soup, or stew until some of the water has evaporated from the liquid. Reduces bulk, often substantially. Concentrates flavour, especially salt, which is why it should not be added until after reduction.
Adding a small knob of butter, while it does not actually thicken your sauce or gravy, will increase the fatty mouth-feel and can make a reduction seem thicker and more pleasing.
i) Too Thick: Add liquid, stir it in thoroughly.
ii) Too Thin:
a) If you used flour: in a small bowl, cup measure, or coffee mug, add cold water very very very slowly to 1/4 C flour, stirring madly, until you have a smooth paste. Add the paste spoonful by spoonful, then bring back to boil to cook and thicken. Adds minimal bulk; dilutes flavour slightly. The flour/butter paste discussed above also works well.
b) If you are desperate for time, or out of flour, toss in a handful of potato flakes and stir them in well. Adds bulk and potato flavour.
c) If you used starch, or don't want to add any bulk or alter or dilute a delicate flavour, use a starch slurry, remembering to bring your dish or sauce back to the boil briefly once the slurry is well stirred in.
iii) Pale Colour/Bland Flavour:
a) If you want to improve colour and flavour, and you have time, brown an onion or two or some mushrooms, or both, and deglaze the pan into your gravy. Use the actual mushrooms and onion for garnish, if you like, or put them in the fridge and use them later.
b) If the colour is fine but the flavour is bland, especially if you are making a reduction or a starch-thickened sauce with very little fat in it, stir in a bit of butter to give both flavour (carried mostly in fats) and thickness (by improving mouth-feel) a swift kick.
c) If the flavour is great but the colour is pale and flabby, and you have time, make a dark roux and stir it in.
If the colour is dark but odd, there may not be a lot you can do. I made a black-bean and chopped-spinach dish the other night that looked, if I am honest, as if the Swamp Thing had had a small digestive upset in the pot, but it was already very rich in flavour and had as much salt from various sources (ham, liquor from canned beans) as I could afford. Sometimes you just can't do much before you run up against Diminishing Returns. If you have a good but greenish dish and you have some canned beets handy you can try beet juice, but I'm not promising a thing.
Consider candlelight. Consider fewer candles. Consider opening more wine. Add a bit of it to the dish, if you want.
d) If you're seriously up against it as far as time goes, sneak in half of a boullion cube, a shot of Bisto or Oxo, or a capful of Worcestershire sauce. These will alter the flavour as well as adding colour, and the boullion, Oxo, and Bisto have a lot of salt: this is one reason why you never want to salt your gravies, stews, or sauces until the last moment, the other being that until you know how much volume you have you don't know how much salt you need. You can also use soy sauce, but be very very careful with it as it can easily take over and drown all the other flavours. If you have dried mushrooms handy, you can use a mortar and pestle to crush them to a powder and add the powder to your dish.
St Lawrence of Rome is the patron saint of cooks. I'm not saying he'll help you out. I'm not saying he won't. I'm just sayin', it can't hurt to know this.
iv) Too Salty: Same as Too Thin, but also add more liquid, and possibly a dollop of unsalted butter, to avoid Too Thick or Better But Now the Flavour's Gone Weirdly Thin. Potato flakes are especially good for sopping up salt.
v) Lumpy: if you have not yet boiled the liquid and cooked the starch, try a whisk, or smashing the lumps with a spoon against the side of the pot or pan. If you have cooked the starch, you are going to need a sieve. And a second pot. And some decent privacy.
Throw everyone you don't need out of the kitchen. Set your second pot or bowl in the sink, with the sieve inside it. Slowly and carefully pour your hot, lumpy gravy or sauce into the sieve, and let it trickle through, stirring gently so that the lumps don't block the sieve holes.
Copious thanks to: themeletor, tenacious_snail, kd5mdk, and fairestcat, for reading, questions, answers, and suggestions.