Those with a passing familiarity with electronic components will, in all likelihood, consider this just plain silly. However, as I have long had an interest in electronics combined with an apparent irrevocable curse when it comes to actually building a working circuit, I found this experience comforting and inspiring.
Central to modern electronics is a device called the transistor. It allows a small current to control a larger one, which essentially means it's an amplifier.
The upper of the two schematics below shows a circuit that doesn't work. Your skin allows small amounts of current to pass, but if you hook your skin to a 5 volt power source, the current coming out of it won't be enough to light an LED. The second schematic, though, is the one I just built, and shows a transistor in action. The fact that it actually worked still makes me grin. For that matter, it also got Andrew fairly excited -- he likes that kind of thing.
The working circuit allows the tiny current from your skin into a transistor. That convinces the transistor to allow lots of current through, which is adequate to light the LED. Better still, the tighter you grab the wires, or the wetter your hands, or in short the better a conductor you are, the brighter the light gets.
Now what I'm not sure of is why this works when I only grab the wire going into the transistor. I don't even have to is greater than my power supply's ground potential. I haven't seen it light up brighter when I shuffle my feet on the carpet; that would be cool :)
Update: Ok, I do know why this works -- the ground potential here is more positive than the ground potential in my power supply, which in turn is based, I guess, on the neutral level at the power plant (my power supply doesn't use the ground plug in the outlet here). So current flows from me and everything else to the power plant even if I don't hold the positive wire.