Something just occurred to me, for the life of me, I cannot remember the Driver's name. I am sure this guy does not have a name for the whole movie. Someone had to call out to him.
I have heard him being referred to as "kid", "guy" and "neighbour".
Did he have a name? And if not what was the purpose of having him without a name? How was Nicolas Winding Refn able to accomplish this trick?
I'll expound a little on what Christian says above as, while I think his answer is correct and sufficient, it was this very same question that led to a deeper understanding of the film for me.
I also wondered about the lack of a name for the driver - until a specific moment in the film which I think explains most of the motivation for what is presented. When the driver asks his benefactor if he's familiar with the story of the scorpion and the frog, all one needs to do is look up the fable (see wikipedia) - it went a long way to answering most questions I had about the film. It also made me realize that his name was not important. I'm reminded of the Shakespeare quote "a rose by any other name would smell as sweet". Stop reading here if you don't want to see spoilers.
The driver's scorpion jacket represents his backstory (almost literally). As Christian alludes to, the title "drive" probably refers more to the character's nature than his vocation. The "kid", or driver, is by nature like the scorpion - a natural killer. The film is likely examining the questions, (1) can one's nature be suppressed? And, (2) can redemption be found by going against nature when one's nature is evil?
The scorpion (the "kid") appears to be willing to suppress his nature and replace it, for a limited time, with the nature of his victim, the frog. The car is the vehicle (sorry) by which the scorpion transforms into the frog and carrying passengers safely to their destination is the scorpion's view of a frog's nature. However, the scorpion is only willing to suppress his own nature for a limited time (5 minutes), after which, he returns to being the scorpion.
The transformation is what allows him to find love with Irene. In most scenes with her - they are either driving together or he is out of his skin (without scorpion jacket). The elevator scene (wearing jacket) is where he reveals his true nature to Irene and she is horrified by it.
The final scenes showing the kid reawakening inside the car and then driving away is perhaps a sign that he has found redemption? Maybe signifying his rebirth as the frog? Pretty cool film.
He did not have a name and a lot of reviewers and critics are comparing him to The Man With No Name as a tough, near dialogue-less man who only has his goal in mind.
As far as how the director achieved this, it seems to me just clever writing not too different to how it was pulled off in Fight Club.
Well, first of all the movie's story is based on James Sallis' novel and he doesn't have a name in there either (it must be even harder for a narrator to call him driver all the time ;)). So in addition to what TylerShads already said, it could partly also just be to do justice to the novel (whereas I'm sure they haven't ported the novel one-to-one, it would be a bit strange from the script writer to come up with another/any name for the main character himself).
I have to admit, I haven't read the novel (not yet maybe?), but from the information I gathered about it (after watching the movie and finding it brilliant), it seems the driver is a much more amoral and emotionless character than in the movie. Whereas his moments with Irene and her family (which have a much larger part in the movie) make him more likeable in the movie. On the other hand the book, although giving some details about his backstory and upbringing, doesn't try to analyse his psychology or provide reasons for his motivations and actions (neither does the movie to some degree). So I think in this case removal of a name helps especially to establish a kind of distance to the character and to make him a kind of black-box whose actions you can observe, but never really understand the driving force (pun intended) inside.
And to accomplishing it, how often do you really call the people you're talking to by name? I think you might wonder how seldom. It's only us movie-trained audience who want to see everybody called by name all the time. Just like TylerShads already said, think of Fight Club (no, his name was not Jack even if everybody tries so hard to establish this convention ;)). And by the way, it was rather Hossein Amini who accomplished it, as this was actually the first of his movies Nicolas Winding Refn didn't write the script himself for, but sorry for the nit-picking.
One of the great things about Drive it that it doesn't pander to normal Hollywood convention.
Some of the key choices reinforce each other. The Driver doesn't have any backstory to explain who he is or why he behaves the way he does. The choice not to give him a name seems deliberate and seems to reinforce his lack of a backstory, making him similar to the Clint Eastwood character in The Fistful of Dollars movies with noble instincts but violent tendencies.
The unresolved ending also reinforces the same atmosphere. The Driver is like a Ronin, a drifting warrior with no master but himself. The audience has to decide for itself who he is and what are his motivations.
