It leverages the fact that the LED is a diode, not a bulb. The circuit uses a transistor and a transformer to send pulses through the LED (about 50,000 per second) to essentially blink the LED instead of lighting it continuously. This allows the LED to light using less than 1V in a AA battery, instead of the normal 3.2V to light a LED.
It's calleda Joule Thief because, when most electronics consider a 1.5V battery to be dead at around 1.2V, this circuit will light an LED down to about 0.8V before the light becomes useless.
-PJ
really neat.