This chainmail pattern is called barrel weave, and this is my current favorite earring style.
What you need:
2 pairs of pliers (I use needlenose)
Jump rings. I use 18 or 19 AWG rings size 5/32 inches (inner diameter). You can make this pattern with bigger rings, but I like it better smaller.
Assuming you make your earrings the same length as mine, you'll need 10 outer rings and 6 inner rings for each earring.
Note: I'm using anodized titanium rings for the colors - check out my other instructable to learn how to anodize the rings yourself: http://www.instructables.com/id/Anodizing-Titanium-Rings/
Note: you can buy jump rings and earring hooks from www.theringlord.com or www.bluebuddhaboutique.com among other places.
Step 1: Preparation
This method of earring construction is designed to minimize opening and closing of the inner rings, because colored rings can sometimes be scratched by the pliers.
I'm assuming you know how to open and close jump rings correctly.
First, close all the inner rings.
Then, open all the outer rings.