So what is a nerd, and how does a nerd compare to a geek? There's a lot of controversy over this, and some people are just wrong. Here's some background on geeks and nerds, and how I'd explain the difference.
The term "nerd" first appeared in print in a 1950 Dr. Suess story If I Ran the Zoo. But no one knows really why he used it, and the has little relevance here... but it's true.
Nerds are more cerebral than the geek, and they generally have abnormally high aptitude for things such as science, technology, and even computers. Nerds, unlike geeks, have marketable skills and a drive to accomplish and achieve. Nerds also can have social skills, play sports, and in general lead fairly normal lives.
The term "geek" originally referred to carnival sideshow wild men who did freakish things such as biting heads off chickens. Geeks aren't quite that strange now, but they're still pretty much useless... they don't have useful skills like nerds do. And they play WoW (World of Warcraft), Dungeons and Dragons, and quote Monty Python... and other stuff like that. Not good... and some carry their calculators in holsters on their belt. Don't ever to that... it's not cool. I knew a guy in school who did that.
Here's a video about geeks. If you have 7 minutes to waste, check it out.
I think these pictures kind of illustrate the difference.