Why does an LED light up when I touch it?
EDIT:

It's been pointed out in the comments that I've phrased this rather misleadingly, let me take another stab at this with the following caveat: while I'm confident I have the following set of interactions correct, but I'm uncertain of what exactly the terminology should be.

As a person, you are largely a radio wave sink.  In addition to the myriad of electron sources I've pointed out below, you're also continually rectifying radio waves into extremely small AC currents/eddies.  In effect: you are a big, really ineffective antenna.

So after the electrons on the surface of your skin which have built up drain off as DC, the difference between yourself and ground will fluctuate in time with the radio waves you rectify.  In the US, this is dominated largely by 60HZ, the frequency of the AC current running to all your sockets.



Original answer:

"Super bright" LEDs or other high-efficiency LEDs are designed to turn a very small amount of electricity into light.  Since the human body acts like a big capacitor, and you exist in an environment awash in (harmless) radio waves and electrons making a mad leap through the air, or being pulled from your clothes or fuzzy carpets: you tend to build up a bit of charge!

In most environments this bleeds off through your contact with less charged substances, and sometimes simply through the less-charged molecules in the air (especially moisture in the form of humidity).

If you give that charge a different route to earth ground, such as through and LED leg, you'll bleed off that charge a bit more quickly and put a small amount of electricity through the leg.  This is enough to get that extremely high-efficiency element to glow very dimly!

Now as for why it work on either leg: there's a threshold of voltage you have to put across a diode to get any electricity through it (silicon is generally about 0.7V at room temperature, gallium arsenide is around 0.3V, etc) and the electricity building up on your skin is AC.  What end up happening is that the LED is getting a tiny AC current around this threshold, and the diode part of the Light Emitting Diode means that the current only flows the "right" way.  Thus, light!

Hope that helps.