I've heard much discussion on these now-common words, so I decided to compile my own definitions. These were copied/summarized/augmented from the comments by Zen Master, Skylar, and especially Beej on this forum.
Dork is a purely derogatory term and carries with it no implication of useful skills or knowledge. A dork will often try one-up others with obscure facts, displaying a total lack of social skills. A lack of personal hygiene can also be associated with a dork. Typically, one does not aspire to be a dork.
Nerd is also a derogatory term, but slightly less so than dork since there is an implication of skills and/or knowledge. However, a useful outlet for those skills is typically missing. Amateur status is also implied. Some can share the lack of personal hygiene associated with dorks, and a lack of fashion sense is nearly certain. Nerds may wish for more popularity than they actually enjoy, but they may exercise upward mobility to geekdom.
Geek is not a derogatory term like nerd and dork (though be careful when using it as some people still believe it to be an insult). It implies hireability due to highly refined skills and/or knowledge. Geeks are marked by an unusual passion and devotion to a subject such as technology or science. Simply put, geeks have made a success of their nerdiness. They are usually comfortable will who they are and have learned to deal with basic social situations. Typical attire is casual though not necessarily trendy.
Linux dork: Posts "Open sources pwns! Micro$soft suXORs!" messages regularly on Slashdot.
Linux nerd: Installs Linux and uses it.
Linux geek: Fixes the bugs, sends bug fix patches to authors, and patches appear in the next version.
Fitness dork: Wears Lycra, but never breaks a sweat.
Fitness nerd: Runs a 5K every now and then, and everybody within earshot hears about it.
Fitness geek: Turns down social events because they conflict with Sunday 8:00 am running schedule.
Science dork: Watches Star Trek and believes the physics is real.
Science nerd: Reads Scientific American.
Science geek: Works in a lab.