This will be the first part of my
chainmaille tutorial; in the second I'll show you how to do one of the more basic weaves. Bear in mind that I've only been making
maille for about a year, so I may do this somewhat wonky (also, it's hard to get a picture of my own hands!).
There aren't many tools that go into making maille, so it's a pretty inexpensive kind of art to get into. Even pre-cut jump rings are pretty darn inexpensive (I use pre-cut aluminum usually- sometimes I'll make my own smaller rings, but usually I just buy them since it's easier). Of course, if you use gold or silver, it would be more expensive of a hobby.
These are the tools I use:

Two flat-nose pliers, a jump ring opener ring, and lots and lots of jump rings.
Here's a close-up of the jump ring opener ring:

And this is how it works:

It's basically just a metal ring that goes over the first knuckles of your finger, and you stick the jump ring into the size slot that fits the jump rings you're using, and use that as leverage for opening the ring with the pliers in your other hand. This comes in handy if you're currently using the hand with the ring to hold the
maille (as in the picture below), or the pliers in that hand are occupied with holding the
maille and you need another ring opened.
When I use the jump ring opener ring, this is usually what my hand looks like, with the second pair of pliers at the ready:

Technically, if you have a jump ring opener ring, you'd only need to have one pair of pliers (since it can serve the purpose of the second), but I find that it's much easier to make
maille with two pairs. The jump ring opener ring isn't even a requirement- I rarely use mine because I'm used to just having two pairs of pliers (I only got the ring opener ring a few weeks ago), but it does come in handy sometimes. Plus, it's a very inexpensive tool (I found mine for about $2 on eBay), so it's probably worth trying out; it may make your
maille working easier.
I'd recommend checking out http://theringlord.com/ if you'd like to get into chainmaille weaving; that's where I get most of my supplies, and everything is great quality. The other useful site I've found is http://www.mailleartisans.org/weaves/ where you can get all sorts of info on how to make the different types of weaves.
Thanks to Susan of MeriGreenleaf for sharing this great tutorial. Stay tuned for Part II, coming soon.