There are many different articles and videos to be found online that will teach you how to build a Joule Thief circuit and power LEDs off of dead batteries. Most of them, however, call for some form of coil made using a ring, or toroid, made of ferrite. This can be rather daunting to newcomers to electronics, as "ferrite toroid" sounds rather impressive and exotic at first.
The truth is that a ferrite toroid coil is just a simple coil with two windings wrapped around a ring made of ferrite (iron). Toroids are often used because the ring shape is very effective at storing electromagnetic energy, but you don't actually need a toroid to make a Joule Thief. A simple coil with two windings around a central core will work just fine.
In reality, ferrite toroids are actually fairly easy to come by, as they can be purchased online from a site like Mouser for less than a dollar or found in the guts of such items as old microwaves and ceiling fan control panels, but just in case your dad won't let you tear the microwave apart and you don't feel like paying shipping on a $0.70 item, I'm going to show you how to make a simple coil for a Joule Thief using a few simple items you can find around the house.
Note: If you have any common sense, you will likely not get hurt in the making or usage of this project. If you do, however, it is you own fault and I accept no responsibility for any harm incurring as a result of your attempt at this project.
The image on the Intro step is not mine. It comes from this video here, which provides a good tutorial on how to make a basic Joule Thief.
Step 1: Materials and Tools
The items you'll need to make this coil are simple:
A length of thin magnet wire. I used about 25 feet of 30 gauge.
A magnetic core. Anything that will stick strongly to a magnet will work. I used a hacksaw to cut the end off of an old drill bit, but a bit of a nail or other iron object will work. You might try the piston from an old sprinkler valve solenoid. Try and make it about a centimeter long, cutting it if you have to (obviously, be careful).
Two buttons. I took these off of a dress shirt. You could also use some little discs or squares of plastic or even carboard if you want. These will serve as endstops to keep the wire wrapped tightly around the core.
Superglue. Shouldn't need to elaborate there.
You might also want to use a bit of adhesive to secure the wire on the coil after you finish winding it. Nail polish topcoat works very well for this. You could also use a dab of silicone or hot glue. I would advise against using superglue, at it will get everywhere and some people claim it eats through the ceramic insulation on the wire.
Needlenose pliers, or small wire pliers. If they don't have a wire cutter built in, you'll need wire snips or a way to cut the wire (I would advise against simply tearing at it like a caveman until it breaks).
Sandpaper. This will be used to strip the ends of the magnet wire. You can also use nail polish remover or a razor blade to do this.
A hacksaw or some other means to cut your core down to size (assuming you need to, of course).
Step 2: Build the Core for Your Coil
To do this, we're essentially just going to glue the buttons onto the ends of the iron core.
Firstly, take a hacksaw, razor knife (be careful!) or wire snips and make a small notch in one of the buttons. This will help when winding the wire around the core.
Next, use a drop of super glue to glue the button onto the end of the core. Try and get it as centered as possible.
When the glue dries, flip the core over and glue the second button to the other side, again making it as centered as you can.
Let the glue dry, and you have your core.
Step 3: Winding the Coil.
To start, cut a piece of magnet wire about 20-25 feet long, and bend it exactly in half so that you have a small loop with two longs ends trailing from it. Be careful, as these will tangle and kink very easily.
Place the loop into the notch you made in one of the buttons on your core, with about two inches sticking out. This will make it easier to wind by keeping the wire from moving.
Wind the two trailing wires side-by-side around the core. Wrap them tightly, moving up and down the core so that they evenly cover the length of the metal with several layers. Try to keep the wires from twisting or kinking to ensure smooth windings (not terribly important to the overall function of the coil, but makes it easier to wind and fix errors and helps prevent the insulation from wearing off the wire, causing a short).
When you have about two inches left of the wires, take your wire snips and cut the loop of wire coming out of the side of the coil in half. You will now have two separate wires wound around a single magnetic core. Twist them gently together in order to secure them.
Finally, use sandpaper, acetone or a razor blade to remove the insulation from the ends of the four wires. You can now use a bit of nail polish or silicone to glue them in place and keep the coil from unraveling.
Step 4: Implementing and Troubleshooting
You now have a coil that will work in a Joule Thief circuit. Use a multimeter or continuity tester to determine and mark which wire end is connected to which and to make sure there isn't a short that has all four wires connected together. If everything checks out, then connect the coil to your Joule Thief using alligator clips, a breadboard, soldering iron or some other method. This should work fine for a simple Joule Thief circuit, but you can always make one with more wire and windings to get more inductance and more power if you need to.
Step 5: Defy Physics and Power Your Devices With Dead Batteries!
You get the idea.