What's the quickest way to get the skin off of an eggplant?

That depends on how you are cooking it. You can just peel it using a vegetable/fruit peeler, but remember to peel towards you to get longer strips. That way you can cook it without the peel in the oven (but remember to slice into it with the knife, especially in the top, otherwise it wont cook evenly. IF you are making something where the eggplant needs no shape to it (i.e. baba ghanouj) you can grill it turning it onto 4 sides, for 10 minutes on a side, and then wrap it in saran wrap like a little candy. Use a scissors to cut off the tip or cap, and then squeeze the eggplant out of the peel (NOTE: This will make the eggplant mushy and better for a dip)

You could take a knife to it, but if you have the money I'd hurry to the nearest Williams-Sonoma or Crate & Barrel store and invest in this new peeler they have that takes the skin off of peppers. It should work very well on eggplant skin, too. I've only used it on peppers, and it works wonders. It'll only set you back about $10. It's just a fancy-schmancy peeler, but it does a much better job on troublesome skins than a regular potato/apple peeler.

It depends on what you want your skinless eggplant to be do you want is soggy after boiling or dry after roasting?. both boiling and roasting is good.

Roast it over gas flame stove until the outside skin is charred. Then, you could just peel them off easily.

I just skin it like a potato, with a knife.

I just use a potato peeler. just use it right away or it might start to get dark.

a veggie peeler.