On the basis that it's me who actually named the Joule Thief, it's about time I actually put up a video showing how to make one.
This video shows a version that is true to the original design sent in by the mysterious Z Kaparnik to the Ingenuity Unlimited section of an electronics magazine called EPE. I just loved the simplicity of the circuit and built many of them. Latterly when the Internet "happened" I added it to my website as a project and gave it the snappy name Joule Thief on the basis that it steals all the last energy from a battery. The combination of the circuit's amazing simplicity, functionality and the fun name just developed a life of its own on the 'net.
There are a few variants on the design which add extra components to improve efficiency, but a true Joule Thief uses a single transistor, 1K resistor, hand wound ferrite bead transformer and the LED you want to light.
The circuit is just incredibly rugged. It can be used with just about any small ferrite toroid or core you can find. It's great for use with the little ferrite rings found in the circuitry of dead compact fluorescent lamps.
The wire you wind it with can be new or salvaged lacquered wire (used in transformers and relays), and the gauge of the wire isn't too critical. The transistor is a general purpose small NPN transistor. You can either use the suggested one or just whatever you can find in your electronic hoard.