This was actually quite fun to build, a quickly gratifying small job: how do you drive an LED from very little voltage (as in, a single 1.5V cell)? You can't do it directly because leds need more voltage (at least 1.6V for reds, above 3.0V for many/most whites).
So you need some booster circuit. Clive has a nice set of instructions for making what he calls a "Joule Thief", a simple inverter with three parts only: a centre-tapped inductor, a resistor and a transistor (He also has articles on other Must-Have Cool Things, like how to make a USB-powered turd).
For the ham-fisted among us, these guys show how to build the same setup with larger-sized parts.
I had a few minutes of nothing better to do this arvo, and built three variants with a fat 10mm white led: one hand-wound largish coil (2cm dia), one salvaged coil of similar size, and one smaller hand-wound one (0.9cm dia) with which the circuit wouldn't light up continuously.
For the adventurous, Dick Cappel has another set of really nice pages on similar projects, like the Rusty Nail LED inverter.