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: https://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.
Step 2: Build the First Unit
First, put one open ring on an earring hook, and close it.
Then, put a second open ring on the ring you just added. Put two closed rings on the open ring, then close the open ring.
Now, put another open ring through the two closed rings. Fold it back on the chain and hook it through the first ring - the one that's attached to the earring hook. (Look at the pictures if this doesn't makes sense.)
Note: when building each unit, it's important that this ring is added in the same direction and hooked through the first ring in the same direction. If not, the units may end up tilted different directions which might look odd in the finished chain.
Finally, add an open ring at the bottom. It should go through the point where the two spread rings overlap. (Again, look at the pictures.)
Step 3: Add More Units
The bottom ring of the first unit is also the top ring of the second unit. Build the second unit the same way as the first one.
My earrings are three units long, but you can make yours however long you like.
Step 4: Experiment With Different Colors
Try out different patterns and colors, and then have fun wearing your earrings.