We've been doing a number of chain mail classes lately (with a new one coming up this Wednesday, July 27, called Mermaid's Tail Earrings), so we figured we'd add a quickie tutorial on how to make jump rings.
First, you'll want to select your coiling tool, which could be a knitting needle, screwdriver, pen, etc. You can get an idea of how big your jump rings will be by measuring the diameter of your coiling tool with a caliper. I'm opting for a knitting needle about 6mm in diameter, which will result in jump rings about that size (the inner diameter will be about 6mm, while the outer will be slightly bigger).
Select a decent gauge of wire (20G or thicker) and start by coiling your wire around your tool. Try to keep your coils nice and tight. Keep coiling until you have about an inch or so of coiled wire. When you're done, just slide your coil off of your tool (a metal tool works best – the coil slides off more easily than on wood, where it can stick if you've coiled too tightly).
Next, you'll use a pair of flush-cutters to begin snipping your jump rings. You always want to keep the flat side of the flush-cutters facing your work, so you'll need to keep flipping the flush-cutters over as you cut, and snipping off the very tip of your wire from the previous cut.
Notice how the cutters are flipped in the next image? Each time you cut, you'll want to flip your cutters so the flat side is facing the jump ring. This is to make sure the cut ends of the jump ring are as even as possible.
However, you may want to file the ends of your jump rings, as this helps to make sure they're as flush as possible – ideally, you want the ends to fit snugly together. But you've just made your first batch of jump rings!
You can find a full tutorial of this process by downloading this .PDF -- it also includes the sawing technique for cutting jump rings. Enjoy!