Byzantine weave isn't actually Byzantine in origin, but it is a beautiful weave and is fairly simple. It's a good beginner's weave for chains (as opposed to sheets) of chainmail.
It is a cousin of the box weave, which I personally dislike, and therefore have not presented as an instructable. If you do know box weave, however, think of the Byzantine as alternating box weave.

Note! This instructable assumes that you have basic chain working skills.
You may wish to start with Paul the Mole's European 4-in-1 weave instructable to get you familiar with chain working. Euro 4-in-1 is excellent for making sheets of chainmail, and is a great beginner weave. I also have an instructable on making jewelry-size rings, and armor-size rings will be addressed in another instructable.

You will need rings and two pairs of pliers. If you're working with small and soft rings, you can use a ring-tool and a single pair of pliers. I work with tooth-less pliers, as teeth will mark my rings. For many ring sizes, you can use chain, flat, bent or needle nosed pliers, but for some sizes you'll need specific ones. Apply judgment as needed.

Ring size: I frequently use 18ga aluminum wire with 3/16" interior diameter as a good medium ring size. You can go with larger or smaller rings, though the wire-to-ID ratio needs to be within a certain range to produce aesthetically pleasing results. A small wire size with large ID will result in a rather... anemic looking weave, and a thick wire with small ID will be too tight to work with.
See the photo? The IDs are all the same, but the thicker wire looks better. Be careful not to go too thick, though--the weave will get too tight to construct.

If you are interested in additional weaves, let me know (though PMs, emails or comments) on what you want me to demonstrate. If you want an idea of what's out there, take a look at the chainmail gallery on my website.

New and shiny updated video on step 3!

Step 1: Prepare Your Rings

I much prefer to prepare my rings before I start weaving. I figure out how many open rings I need and now many closed rings (I calculate a ratio), and then I start weaving. When I run out, I prepare more rings in the appropriate ratio.

For this method of Byzantine weave, you want 4 open rings per 2 closed rings. These rings will be made into 1 unit of 2x2 for every 2 open rings. Let me explain:
Take one open ring and slip two closed rings on it. Close the open ring.
Take a second open ring, and slip the same two closed rings upon it. Close the open ring.
You should end up with a set of four rings where every ring goes through two other rings, no more and no less.

Once you prepare your 4:2 ratio of rings, you will be left with 2 open rings per 2x2 unit.

Step 2: Start Attaching the 2x2 Units

Take an open ring. You can either anchor it to something, or you can treat it as your source ring.

Before you close the ring, slip a 2x2 unit on to it. Now close the ring.

Take a second open ring, and slip it in place parallel to the first open (but now closed) ring. Make sure it goes through exactly the same rings that the first open ring went through, but that the first open ring does not cross the new open ring.

Call the pair of open (but now closed) rings "pair A."
The next pair of rings will be "pair B" and the outer most rings are "pair C."

Step 3: Elongate the Chain

Take the two outer most rings (pair C) and flop them so that they are perpendicular to the next pair of rings (pair B) down.

You will attach your two rings to pair C, but pair B will flank your connecting ring: see photo! The open/connecting rings you attach in this step can be called pair D, but you'll see that they're actually a second iteration of pair A.
Before you close this new ring, put on a 2x2 unit! Now you can close the ring.
Now give it a partner ring. Make sure that this newest open ring goes through pair C, and the 2x2 unit as well.

This step sound really difficult and convoluted, but once you actually hook your ring onto pair C (between the legs of pair B), things fall into place. I promise.

But now, all you need to do to continue is to flop the outer most pair of rings on your newest 2x2 unit, use two open rings and attach another unit of 2x2!

New Video! Now with color coded rings! And new background music!
Alright, here's a break down of what's happening in the video clip v.2:
- 2x2 units are constructed 2 copper + 2 gold. (It's hard to tell the color, but I consistently place new 2x2s so that the gold rings are the ones sticking out at the end.)
- Open rings that will be used as connectors are black.
- Black/open rings connect copper to gold.
- I point out where the black ring attaches to the gold rings by use of the bright blue twist-tie.

Whenever I finish adding a 2x2, the rings at the end are gold. My next open (black) ring hooks on to these gold rings, but watch how the rings have to be flipped first and then attached from between the copper rings of the 2x2 unit. (Or refer to photo set.)

Rings are 14g 3/8" bronze and anodized aluminum. The background music is Jack Hinks, and later Trois navires de ble by Great Big Sea.

Old Video (I don't know if it's fully supplanted by v.2, but it's still here in case you can learn something else from it.)

Rings are 18g 3/16" bronze. The background music is Great Big Sea playing the song "French Perfume" (album: Sea of No Cares)

I'm not usually this fumbly when I weave, but there's a camera in my face, and I have to work with the chain held out further than normal so it just looks really clumsy.

Step 4: Stopping the Chain

You have a few options, depending on what you're trying to make.

The easiest way might be to stop at "pair A" and simply not add the next 2x2 unit.
You can actually use something else in place of pair A: a split-ring for holding keys, or perhaps a different ring size/type/something.

If you think you understand the weave well enough, you can also bring the end of the chain to the beginning and attach it in a circle.

It is also possible to attach the end to a point somewhere else on the chain, too, to form a loop at one end (or both).
Simply take pair A and attach to any other pair A prior on the chain.

Step 5: Epilogue, Rosettes & Speedier Speed Weaving

Note! The photos show 16ga bronze 3/16" rings. I really don't advise it--the rings were very stiff to work with in close quarters. The video shows 18ga bronze 3/16".

If you use a looser ring, you can actually prepare 2x2x2x2x2 units! (To prepare for those, have opened/closed rings in a 4:6 ratio.)
To attach these you need a pair of open rings, and you need to do the bizarre flopping thing before you hook the open ring onto your 2x2x2x2x2 unit. Close the ring, and give it a partner ring.
Now you grab the final rings on the 2x2x2x2x2 unit, and flop 'em. Grab an open ring, hook it on (just like step 3), toss on a new 2x2x2x2x2 unit. Close the ring, give it a partner and... repeat.

Rosettes (or "mobius balls") are a great embellishment to use with Byzantine.
When you attach open rings, just make sure to cross them.
Thread the second open ring through the first, you'd get a (slightly wimpy) rosette. You'd best add a third ring, making sure that it passes through the previous two. And possibly a fourth, fifth, etc.
I find that 3-4 is the optimal number, and I usually attach rosettes between units of 2x2x2x2x2 (rather than units of 2x2), so that they're not overwhelming the weave.
You can use a different ring size, color, material, etc. with your rosettes!

Incidentally, you can use a single ring as the connector. This looks better when the single ring is thicker and/or larger than the rings in the 2x2 units.

Step 6: A Few Useful Links

Supplies:
The Ring Lord - Hands down, this is the best supplier of rings. You'll wait a little longer for your rings to ship from Canada, but the price and selection are unbeatable.
UrbanMaille - They specialize in sterling silver rings (even argentium!), and there's a selection of a few other materials. The prices are higher than The Ring Lord's, but if you're working with precious metals, you might as well spend the extra money for quality (the polish is amazing). This is also where I get most of my tools. (Ring tool!)
Rings & Things - This is where I get wire to coil my own rings. The prices are very good, and they carry argentium. (Please note the $25 minimum, and surcharge for all orders under $50. I rarely have trouble meeting the 50$ mark...)

Weave Instructions:
Derakon's Library - Very clear instructions for many weaves. (This was the best instructional site back when I was first starting chain working, and it's still my primary reference site.)
CG Maille - (Previously Phong's Chainmaille Tutorials) Computer generated graphics for the tutorials, covers the same range as Derakon, but the graphics are prettier. So shiny.

Forums:
The Ring Lord's Forum - Very active forum for chainmail (it might well be the most active out there).
M.A.I.L. - An active gathering of maillers, and it's much more than just a forum. There are instructions for weaves, articles on chainmail related subjects, galleries... the list goes on. It's good because any member can submit his/her version or interpretation a weave or theory. That also means you may have to wade through some pages that are of poorer quality.

Other Information: (Probably overkill for people just beginning to weave, though.)
Zlock's Aspect Ratio Pages - This appears to be the wellspring of in-depth information on aspect ratios. There's a handy chart,
Venom's Pit - There are a couple of charts that you might find handy, once you really get into working with chainmail.

There are many, many more sites out there that I have not listed. I have pulled what I believe are the best, and I intend to keep the listing short. However, if you violently disagree, or believe that I've grossly overlooked another site, feel free to let me know.