Today at the store I noticed a setting on the grinder for Turkish coffee. It was the most fine grind of coffee possible. (At least on that grinder. )
Is there some special method for brewing this?
What (if anything?) would happen if I used a Turkish ground coffee in my regular old drip coffee maker? Would the end product be stronger? (In taste and/or color?) Would it contain now more caffeine?

Turkish coffee is prepared by boiling finely powdered roast coffee beans in a pot, possibly with sugar, and serving it into a cup, where the dregs settle.
That’s pretty much how cowboys made their coffee, except that they didn’t grind it so finely. They would toss some eggshell or some cold water into the pot to settle the grinds.
If you use a turkish grind in a regular coffee maker, it would clog the filter, and you’d have a godawful mess to deal with. I don’t know if you could use a french press or not, but I’d suggest just doing what the Turks do.
The same coffee is also known as Greek, Arabic, Armenian, Albanian, or Bosnian coffee, by the way.
Having been ground more finely, the coffee will release more of everything that it releases – color, flavor, and caffeine.
Whatever you do, DO NOT BREW IT IN YOUR REGULAR BREWER, you will have a BIG mess. The fine grind will clog the pores of the brew basket and the result is overflow with grounds and HOT coffee everywhere!
I had to get that out of the way first so to avoid a potentially hazardous situation. To brew it mix a few TBS in a cup mixed up with boiling water and allowed to sit for a short time to allow the mixture to brew, and the grounds to settle. You drink it as is, grounds and all. Not my thing, but you might like the added crunchies at the bottom of your cup. It will provide you with a very stimulating jolt, those with high blood pressure, or on certain medication should not consume this amount of caffeine all at once.