What you need:
Flat nose Pliers - just to help with the final piece around the nail
A Nail - or anything that is the size of the Jump Rings you want to make
A length of wire - I use 0.8mm Silver plated Copper here
Wrap the wire around the nail until all the wire is used up. Wrap it as tightly together as possible as gaps result in oval and mis-shapen rings.
Remove the coil from the nail.
Starting at one end of the coil you need to cut where the wire is just meeting the next coil so when you cut here you will have a full ring that will be slightly open.
Now you can use your jump rings you just open them further or close them up.
Jump Rings are needed for practically everything you do in jewellery making and will never go to waste so always make a few extra and keep them stored. You don't want to have to make loads of them when you only need 1 or 2.