In sanscrit, the term Avatar comes to mean “the terrestrial incarnation of a god”. In nerd, it is an image that you use to represent yourself in forums or while chatting, usually a picture of some person or character you feel attached to or a caricature of yourself.
Besides searching for a premade picture, you can make use of one of the many avilable automated generators, from which I chose the two examples above, to tailor one to your liking. The Gaia Dream Creator offers you full-body manga style characters with lots of accesories and a high cuteness factor; the Avatar Creator (which I found in numerous sites) is a flash application focused on face portraits. Both of them are very intuitive to use, and the Gaia Creator even outputs a rendered image that automatically gets hosted in ImageShack for you to use; as for the Avatar Creator, you´ll need to make a screen capture (printscreen button), then paste and cut the portrait with an appropriate program. Have fun!
[Update|26.12.05] If you liked the Dream Creator, you may want to try gavsim too.