won't stay running after taking jumper cables off

The connections to your battery might be damaged/loose or your battery might be completely dead. I'd try taking the clamps off the battery (one at a time) and clean the battery terminals and clamps with a wire brush. The connection might be bad and not allowing current to flow through the battery. The only real way to test your battery is to hook up a load tester like this, or you can take it to a place like Sears auto center for them to hook it up to a testing machine. That'll let you know if you need a new battery or not.