I don't have any formal training as a chef. Like most of you I've learned as I went along. I do have a technical bent (my prior career was as a computer programmer and then editor of programming magazines and books) so I've always been interested in the science and technology of cooking and food. But sometimes it took awhile for me to discover things on my own and the multitude of ways to thicken sauces is one of them.
I grew up in the South and flour was the primary thickener - often made into roux. Cornstarch was a secondary thickener, but I never cared for it (until I learned when to use it) and so I mostly stuck with flour for years. Eventually my reading led me to other thickening options. I not only learned when cornstarch was a good choice, but also how to "mount" butter, using nuts and potatoes, and one of my favorites: arrowroot.
Thickeners are starches. The starch molecules absorb liquid, swell up, and stick together. That's the purpose of a thickener. Flour can complicate this because it also contains gluten - proteins that form long strands before glomming on to each other and one of the reasons flour tends to get lumpy. Coating flour with oil reduces this tendency.Wheat Flour: Mix flour with water and it forms strands of gluten - bread. Mix a lot less flour with some lipid (oil, bacon grease, duck fat), cook it and you get roux. Add roux to a lot of liquid and you get gravy. The trick is the oil or fat coats the particles of flour and slows the formation of gluten, making the sauce controllable. You can also mix flour and room-temp butter together to make beurre manie and keep a jar in your refrigerator. Add beurre manie to a soup or stew to thicken it up in the last five minutes. When thickening with flour cook it at least 5 minutes to eliminate any raw flavors, give the gluten time to form and the starch time to swell.
Corn Starch: Corn starch thickens quickly (as opposed to flour). It also so produces a glossy somewhat gelatinous texture. This is a desired characteristic in many oriental dishes, but I don't care for it in fruits pies and gravies. One advantage cornstarch offers is that it's almost flavorless. Another advantage is that unlike flour-thickened sauces it doesn't separate when frozen. When making a sauce, corn starch is best mixed into a bit of cold water to make a slurry then whisked into the hot liquid.
Tapioca: This has become my standard thickener for desserts like pies and crsips. But the standard Minute tapioca results in funky little bubbles of tapioca - a texture that interferes with the pie. So I grind the tapioca to a fine powder in a coffee grinder and it completely disappears in the pie, the pies' juices remain clear (flour produces cloudy juices) and it's not gelatinous like corn starch. (Tapioca is a bit weird in savory sauces though.) I simply sprinkle the ground tapioca on the fruit and stir it in with any sugar and spices.
Arrowroot: I've become a huge fan of arrowroot (available in the spice section at your supermarket). It's completely tasteless, has no effect on the texture or appearance of a sauce, has tremendous and almost instant thickening power and won't separate when frozen. It's a bit expensive so I use it primarily when I want to freeze a sauce or gravy I would ordinarily use flour in. Like corn starch I add this as a slurry to hot liquid.
Butter: Adding butter to a sauce to thicken it is called "mounting" butter. A corruption of the French term beurre montè refers to whisking pieces of very cold butter into a pan sauce to thicken it. The butter is briefly emulsified (suspended in liquid) without quite melting. It works best in pan sauces served immediately.
Potato starch: You can buy powdered potato starch, but I usually use it to thicken soups and stews that already include potatoes (or wouldn't be hurt by a bit of potato). The next time you make stew, instead of dredging the meat in flour before browning it, just brown it and then pull out a potato, puree it and add it back to the stew. You'll not only et a thicker sauce, but a more deeply flavored sauce. Again, add as a slurry.
Nut flour: This one I discovered on my own - although it is known. I was making a sauce for a lamb dish and didn't want to go with any of the standard thickeners and it occurred to me that nuts are full of starch. So I ground some pine nuts up to make pine nut flour and thickened the sauce with that. Nut flour is a weak thickener, but sometimes that's all you need and the flavor it brings is a bonus. Since then I've used ground almonds, pecans, and hazelnuts as thickeners. The flour is fairly coarse so I whisk it in directly.
These aren't the only things used for thickening. Eggs are common as it a simple reduction which evaporates liquids leaving solids behind. And one of my favorite tricks for thickening a stew or pot roast is to remove a cooked potato, whip it, and stir the puree back in.