The idea and circuit came from this Make weekend project. Why don't you pay them a visit?
Step 1: Parts and Tools
But for those of you who like it in text, here it is:
Helping Hands (Optional)
A Blue or White LED (Other colors are fine, too)
2N3904 Transistor or equivalent
1k Resistor (Brown-Black-Red)
Thin wire, two colors (magnet wire works)
You can get the toroid and transistor from a dead CFL; the transistor is usually labeled 13002.
Also, if you use a 2N4401 or BC337 transistor, your LED will be brighter because they can handle more amps.
Step 2: Wind the Toroid
You can find toroids in a few places. Old computer motherboards, XBOX and X360 motherboards have them (don't take them unless it's dead!). You can find toroids in computer power supplies, or you could buy them at your nearest RadioShack.
Take your two strands of wire, and twist the ends together. You don't have to do this, but it makes winding a little easier.
Thread the twisted end through the toroid, then take the other two ends (Not twisted together) and wind it once around the toroid. Don't twist the wires; make sure that two wires of the same color are not right next to each other.
Keep winding, making sure you wind the coils tight. It will still work if they are kind of loose, but it is better to have them tight.
Ideally, you want about 8-11 turns on your toroid. Even if you can fit more, don't put more on. Make sure the turns are spaced evenly around the toroid.
Once you wind around the whole toroid, cut off the extra wire, making sure you leave a couple of inches for soldering.
Strip some insulation off the wires, then take a wire from each side, making sure they are of the OPPOSITE COLOR. Twist them together, and then you're done with the toroid.
Step 3: Solder It All Together
First, take the two outer leads of the transistor and bend them outwards a little ways, and bend the middle lead backwards. Bend the LED's leads outwards, too. You don't have to, but it makes it easier to solder.
Take one of the wires coming from the toroid that is alone; that is, not twisted together with another wire. Solder it to one side of the resistor. Solder the other end of the resistor to the middle lead of the transistor.
Take the other single wire from the toroid, and solder it to the collector of the transistor. Solder the positive side of the LED to the collector as well, and solder the negative side to the emitter.
All that's left to do now is solder an extension wire to the negative side of the LED. Take a piece of that wire you had earlier and solder it to the transistor's emitter.