A scorpion is visible on the drivers jacket throughout the whole film. He even asks Albert Brooks' character "Did you ever hear the story of the scorpion and the frog?".
What does the scorpion represent?
You talk about the idea of Drive as a superhero film, can you talk about where the idea for the scorpion symbol on Ryan’s jacket came from and the allusion to the scorpion and the frog story within the film?
I knew I wanted a satin jacket because I knew I wanted him to shine at night.
When you deal with very good actors, you often let them figure out their wardrobe themselves because that’s very much how they build their characters, through wardrobe. So Ryan would go out and find a jacket that he’d feel comfortable in and he’d bring it back and we’d make it satin. A lot of them were old military jackets with symbols of eagles and other iconic American symbols, and I thought “oh that would be great if he had a symbol on himself, like a logo!”
We decided we were going to put some kind of animal on the jacket, but then by coincidence I was with the costume designer in a garage looking at how mechanics dress.
Ryan was there working on his car because he was building his own car at the time, and he said that I should show the costume designer my visual reference for the Drive, which was Scorpio Rising. The film starts with the logo of a scorpion, and the telekinetic minds of myself and Ryan simultaneously said “it’s a scorpion!”
Then Ryan had the idea that we could use this design to incorporate this story about the scorpion and the frog, and it turned out really great.
While Oliver_C's answer explains where the idea came from, it doesn't actually explain what the scorpion represents.
One could reasonably interpret the meaning of the scorpion as crime. This can be further extrapolated to also mean that The Driver is The Frog.
In the story The Scorpion and the Frog:
The Frog carries The Scorpion across the stream on its back.
The Scorpion assures The Frog that he wont sting them. The Frog is hesitant, but ultimately agrees to carry him.
The Scorpion stings The Frog while crossing the stream. The Scorpion says it's 'in his nature', and they both die.
the thugs attempt to kill him at the motel. Likewise his predicament worsens as Irene and Benicio are now threatened and he has to continue dealing with the criminals.
Bernie stabs The Driver. Likewise, The Driver is wise to what Bernie will do (since he sees Bernie as The Scorpion) and stabs him back.
are perceived as dead during the agonizing shot of The Driver staring into the distance.
And of course, The Driver is literally carrying an image of a scorpion on his back, further suggesting that he is The Frog.
One could also interpret that The Scorpion could represent Bernie or The Drivers burden of associating with criminals. As noted in Napoleon Wilson's comment to the question, this answer as a lot of analysis on The Drivers character, and submits that he is both the Scorpion and the Frog.