How can I efficiently make the loops for chain mail or small chain crafts?
When I took jewelry making classes, we used the saw method mentioned in Kellerra's answer section "the slow way". 

You start by wrapping your desired wire around something round like a dowel to make a "spring" of circles and then you cut them apart with a saw.

The best setup I found was to have something... either a dowel or steel bar... with a notch cut in the end as a "guide" for your saw blade with the far end held in a giant clamp attached to a table. Here's an image I found that shows what I'm talking about:



Use tension in your off-hand to pull the rings towards the blade notch... note that as you cut through each ring, it will drop, so you want something to catch them unless you want to spend an hour finding where they rolled on the floor.

The saws are pretty easy to use once you get the hang of them... but you will probably break tons of blades when you first start out because you're going to have to figure out the right amount of pressure to use that will cut the metal without snapping the blade.

And, once you get used to the saw, this is actually pretty fast. Granted, it's not as fast as some of the machined methods which, if you're really that interested in speed and you can afford it, are a good option... as is simply buying the rings premade, which isn't as expensive as you might fear if you buy in bulk, though it is more expensive than buying the wire alone.

I think you'll find the cuts to be quite a bit cleaner with this method than with a lot of other methods around, particularly using wire cutters.