basically a gravy is just "thickened liquid" - setting aside the multitude of "chemical" thickeners, traditionally a "starch" is used. the starch can be wheat flour, corn starch, arrow root, potato flour, rice flour, etc. each has its own "flavor profile"
what happens is the starch absorbs water and swells up - the 'solution' gets thick.
couple of things in the background:
if the particles of starch are not "fully dispersed" you get lumps. my mom had a special "shaker can" to "mix" flour and water, then added to the liquid; my childhood gravies were complete with lumps..... stuff like cornstarch is already fine enough that "shaking" can be reduced to "stirring"
next, the "absorbs water and swells up" bit does not happen at room temperature - as mentioned multiple times, the 'schufft' won't reach its "done getting thick" stage until the gravy / sauce has boiled / simmered.
wheat flour, in particular, can carry through a "raw flour" taste if not pre-cooked or cooked "as a gravy" - which explains the invention of "the roux" - typically 1:1 flour and fat - cooked a couple minutes to avoid the "raw flour" taste. personally I find the "cook the raw flour in the gravy" approach to be difficult to control / judge - so I don't do raw flour into hot liquids.
light colored flour roux (white / blond) do indeed have more thickening power than darker roux - the color comes from the toasting effect on the wheat flour and the more it's toasted, the less "swelling" it does. for an equal quantity, the darkest roux aka "brick roux" - I'd guesstimate has about 1/3 the thickening power.
why the roux thing?.... the fat separates the starch particles so you don't get lumps. further along, the darker one have their own toasted, nutty flavor.
with regard to the amount of fat to flour to liquid, that's a very dicey equation. if you're working with a fat free broth, mebbe "guidelines" will pan out. if you're working with pan drippings - well - they will have water, fats and solids - so "a cup" may not quite be the effect as "a cup" of broth.
so what's a cook to do?
for brown gravies I make up a dark roux during a quiet moment and freeze it. after it has cooled I roll it into a "log" and it's a simple matter to pull the roux log out of the freezer, lop off whatever I need and add it to the hot liquid. not thick enough, lop off some more and stir it in. judge the final thickness at a "whole pot" simmer - also keep in mind, as it cools, it gets thicker - so the hot gravy in the pot is "thinner" than the (cooled) gravy in the gravy boat....
for a light sauce/gravy I simply make a roux on the spot - add whatever "pan juice" is available, if it turns out too thick or the voumes is not sufficient to feed the hungry horde, use milk / cream / water / wine / stock for any additional liquid need to adjust consistency. and yes, sometimes you make a bit too much roux and it gets tossed.
cornstarch / arrowroot - easier approach. stir a couple teapsoons into cold water, add to hot liquid, observe thickness state, add more as needed. observe the "full pot simmer" caution. as before, sometimes you have left over starch&water and it gets tossed.
corn starch is a favorite for stir fry methods - it's fast. stir fry the ingredients, recombine with all liquid, add liquid to get the right sauce volume, dribble in the corn starch slurry to achieve the thickness.
observe that corn starch and cold water is not a "solution" - it's a "suspension" - the cornstarch will settle to the bottom. always stir it up prior to dribbling.....