Well there are a few ways to make it. You can buy those pre-made links they sell on E-bay and various other sites, which come in either riveted or butted.
The riveted links have little flat spots on the ends of the links, with a hole through it, and you pass a rivet through that hole, and seals the ring.
You can also buy butted links. Butted links are where the ends are open, and when you just push them shut, and they stay together from the tension of the metal.
There are also people who weld the ends of their links together, I have never done this since I can't really get a hold of a welder for long periods of time. I'm not sure how authentic this is, I know there was pressure welding or whatever it's called in medieval times where two pieces of metal were "welded" together by applying heat and pressure, but I'm not sure if this was ever used really, I'm no expert.
LInks that have connected ends are stronger, since the links can't just pull apart as easily, but I really didn't want to get involved in riveted mail. Plus, IIRC, only royalty or very wealthy people could even afford the riveted stuff, and most common soilders just had butted mail.
I chose to use Butted links for a few reasons. First of all, They are much easier to make, and I would assume it doesn't take as long to make a sheet of butted mail as it does a sheet of riveted mail ( I don't know personally, I've never made the riveted stuff)
To make the links, you can either set up some sort of jig, or just get a metal or wood dowel the same diameter you want (I don't know if there is a specific size you are supposed to use, so I just used what was handy) and then wrap wire around the dowel in a tight coil pattern. When that's all done, take the coil off, and cut the links, I used some heavy duty metal snips.
I'm not sure of the exact size of wire to use, somewhere around .05 is good I think. I really don't know the size of the wire I used, but I think it may have just been a tad bit too thin (although it does hold up to the claw end of a hammer VERY well, haha)
To make a jig, which is what I'm going to do, all you really need is some various pieces of wood, I already have the one I'm going to make out on paper. It's just a little dowel between two peies of vertical wood on each end, with a long peice of wood at the base. The dowel extends about 4 inches past the pieces of wood that holds it up on the right side, then I will drill a hole through it, and put a smaller dowel in that. Then, drill a hole about the size of the wire on the other end, and that is what will secure the wire to the dowel. Then, just turn the dowel by the little crossbar on the other side (like on a vise) and that wire gets coiled onto the dowel. I'm sure there are more complicated methods out there, but it's really all I need for making stuff for personal use.
About the actual pattern of the links, there are some guides out there for showing you the general pattern (there are various different ways of making it) which you could probably find by searching on google. I wish I would have seen these BEFORE I started making mine, it took my a long time just to figure out the pattern, one of those guides really would have helped.