The Joule Thief is a clever little circuit that can light a LED with a battery that is nearly dead. It does this with a pair of opposing magnetic fields. I like this circuit because it is simple to build, yet demonstrates some very complex electrical behavior.
I've designed a printed circuit board to help make this easier to build for beginning Makers. I used open source KiCAD software to lay it out (a future Instructable?), and OSHPark to manufacture it. If you want to just wire one up without the board, check out Angelo's Instructable which uses essentially the same circuit.
If you want to learn a bit more about the principles that make this circuit work, check out this video: http://youtu.be/0GVLnyTdqkg. And, if you want to see some serious research in to the operation and efficiency of the circuit, check out http://cs.yrex.com/ke3fl/htm/JouleThief/JouleThief.htm.
I've seen a number of different circuits using an array of parts, so it is certainly possible to do this with what you have on hand. Here is what I'm using:
Remember, I've done all the hard work if you just want to buy the kit at Tindie.com.
For tools, you will need: