Making the Basic 4-in-1 Pattern
Rings to Mail
The following sequence takes you from coil to chainmail in step-by-step fashion.
Wind the coil

The first step is winding the coil. You can use any method you like, as outlined in the Coil Winding Tools section. This picture just shows a short coil, for demonstration purposes. In practice, your coils will probably be at least a couple of feet long.
Remove the coil from the rod

This can be a bit tricky if, as in the left picture above, you neglected to leave a little gap between the wire and the hole at the beginning of the coil. I usually bend the wire over a bit before I start, leaving a small exposed loop (right image) so I can fit the jaws of the cutters in and cut it off the form easily.
Cut the individual rings apart

The next step can be done any number of ways. As I mentioned in the Cutting Tools Section, I use aviation snips to cut my rings apart. However you choose to do the cutting, you should end up with rings with relatively smooth ends that can be closed back up again (see this picture for an idea of how you want the rings to close back up). They don't need to be perfect, but you don't want mangled ends, either.
Closing the rings

Once you have cut your rings apart, you want to close them. Actually, you'll probably want to leave some open for the next few steps, but you'll probably want to close about two thirds of them. For the purposes of this demo, a couple dozen closed rings will do fine. When working with the rings, you'll notice that the wire will have some tendency to spring back to its original form. This will vary on any number of variables, but you will eventually get a feel for it. You want the rings to close up as completely as possible, with no gaps between the ends. Believe me, you'll know it when a ragged end scratches across your skin. You'll want them to look something like the "closed rings" in this picture.
Making 4-in-1s (optional)

Here we make the basic unit of this particular pattern, the 4-in-1. It's just one ring passing through the center of 4 additional rings. To make it, take one open ring, feed four closed rings onto it, and close it. The end result should look something like the pieces in the picture above.
The reason why this step is optional is that in the next step we will be joining the 4-in-1s together to make a chain. The chain can just be built up as you go, but some people prefer to store their working stock as loose 4-in-1s instead.
1-2 chain

As mentioned before, the 1-2 chain (or "running chain") is just a series of 4-in-1s linked together end to end. It's called 1-2 chain because it alternates between having one and two links together.
I find that this is a good way to keep working stock, as you can just coil the chain up on an old empty wire spool. My friend disagrees, preferring to store his stock as loose 4-in-1s in large buckets. The main difference here is that when our working stock is accidentally knocked off the table, he has to spend fifteen minutes picking up 4-in-1s, while I just have to wind a few feet of chain back onto the spool.
Laying Out The Chains

Start with two short lengths of running chain (with double-links at the ends, as shown), and some open rings. You'll notice that the running chain doesn't lie perfectly flat. This is good. Orient the strands so that the "high end" of the double links is facing away from you, as shown in the picture.
Linking The Chains

This is where things get interesting, going from just a bunch of rings and chains, to becoming chainmail. Link the two chains together with the loose rings from the previous step. Work one ring at a time, from one end towards the other. It's a bit difficult to see from these pictures, so you can click here for a more detailed look and explanation. If you can successfully complete this step, you can make chainmail.
Keep Going

Once you've linked those two chains together, add another chain the same length to the bottom of them. Add another to the bottom of this, and so on. You're making chainmail!
The result of all these chains going together is what I call "ribbon stock", or "4-wide ribbon". This is slao easy and convenient to store on empty wire spools, or you can simply wind it up on itself. From here, lengths of ribbon stock are stitched together to make other things, such as shirts, coifs, and leggings. To connect ribbons together, just line up the patterns as with the chains, and add rings to make the pattern continuous.
And that's all there is to it! Now you're ready to actually make something. I've got a few patterns available to get you started.