I've always understood a nerd to be someone who is book smart, into school, perhaps smarter about books than social stuff, but not necessarily bad at socializing. Nerds are usually introverts. (By this definition I am, and always have been, a total nerd.)
To be a geek, you have to be a geek about something. Many geeks are into computers/technology/video games, but you can also be a film geek or a theater geek or an art geek or whatever. The point is that you're really into something, on a level that most people aren't. Geeks are not necessarily introverted, but if they get too obsessive about whatever they're into, they can have trouble making friends with non-geeks who are turned off by their singular interest(s). (By this definition I'm an art geek, perhaps.)
I know you didn't ask about dorks, but I'll go ahead and give my two cents on that as well. A dork is someone who either isn't aware of or doesn't care about social norms. They don't act the way they're supposed to in social situations. A lot of times these people are class clowns, and have tons of friends. Other times they're just socially awkward.