What are the (most common) different ways to thicken sauces and gravies?
Major ways of thickening a sauce -- or, for that matter, some other liquid:


Starch (e.g., flour, cornstarch, tapioca, arrowroot)
Emulsion (e.g., butter, cream, yolk, mustard, blending in lecithin-and-fat)
Coagulated protein (e.g., blood, egg, pureed seafood)
Gels/polysaccharides (gelatin, alginate, xanthan, pectin, agar agar, guar gum)
Suspension (general puree of solids)


Note that some of the examples of one category can also fit into another.