How do different types of thickener actually thicken?

There are a number of different ways in which gelling agents are classified. Off the top of my head:

• Viscosity (firmness/thickness) of solution and gel forms
• Thermoreversible/irreversible (does it "melt"?)
• Hysteresis (water loss)
• Hydration, melting, and setting points
• Appearance (in particular transparency)
• Sensitivity to heat, cold, alcohol, and pH

Starches are all very similar in this regard. They all form viscous solutions and gelatinize to a stiff and elastic consistency. They are all opaque-to-translucent, have very slow hysteresis, hydrate in cold water and gelatinize in hot water. They tend to tolerate some alcohol and a fairly low pH.

Tapioca has the lowest gelation point, and the most heat stability, and potato starch is similar. Corn starch is particularly weak against acid, heat, and cold (especially freezing), unless it's been modified. There's also arrowroot which is somewhere in between.

Wheat flour is something completely different because it has proteins which form gluten in the presence of water and heat. I wouldn't even consider comparing it to the others, it's sometimes used in gravy but generally you would only use flour to make doughs or batters.

As far as gelatin and all of the other E-numbers are concerned, you can find a lot more about their properties in Kymos' Texture and of course, Modernist Cuisine if you want to shell out the cash. There's quite a lot to read about.

Gelatin, specifically, is thermoreversible, hydrates (blooms) in cold water, disperses in hot water, sets below 15° C, melts at 25-40° C (mouth temperature), tolerates alcohol but is weak against all of the other usual inhibitors, and forms a soft, transparent gel. It's nothing like starches but is a lot like a lot of the other E numbers (Gellan, Agar, Carrageenan, Pectin, etc.) Refer to the aforementioned link for more information.

I thicken gravy with either 2Tbl wheat flour to 1 c of broth, or 1 1/2 Tbl cornstarch to 1 c broth, or else 2 Tbl wheat flour plus 1 1/2 Tbl cornstarch to 2 c broth. In each case, I whisk the thickener into the cold liquid before heating and stirring. I learned to do that, without making a roux, years ago when I had to limit my fat intake. The combination makes the very best texture and flavor.