We've been called all three, and to be honest, we always assumed they meant the same thing. However, according to the cool kids, there are differences.
Official definitions for nerd, geek, and dork each use the words "inept" and "foolish." Nerds have the added distinction of being "unattractive." Ouch. While it's hard to argue with the dictionary, we sought out definitions from the Internet at large.
According to Whatis.com, nerds are people of above-average intelligence who place little importance on their appearance. Nerds are often aware of their status, but they don't mind. In fact, many take pride in the putdown, as it means they're smart and not wrapped up in superficial worries.
Geek is a more specific term. Back in the day, geeks worked at carnivals, and (according to the dictionary) "bit the heads off live chickens." Thankfully, the term now has a different connotation. Like nerds, geeks are smart, but they tend to focus more on technology. As Urban Dictionary explains, these are the people you make fun of in high school and later work for as an adult.
Being called a "dork" is the biggest insult of the three. There's no way you can spin it into something positive. After all, even the dictionary writes that dorks are "stupid" people. And to make matters worse, dorks assume they're cool. Oh, and they smell, too.
So, to sum things up, if someone calls you a geek or a nerd, thank them. If someone calls you a dork, consider going back to school and investing in some new deodorant