When vampires are changed from humans, they acquire certain superhuman abilities; speed is one of these. Edward is capable of moving so quickly that no one spots him at all. The incident where Bella was about to be crushed by the truck is one such example of Edward moving in this way: "All I know is that you weren’t anywhere near me. Tyler didn’t see you either." (TW3)
It isn’t until chapter 13 that we see the true nature of this speed, where the distance covered in several hours of hiking is retracted in a matter of moments, with Edward carrying Bella on his back. The speed with which he races is shown to be entirely effortless, and is pure exhilarating enjoyment: "He streaked through the dark, thick underbrush of the forest, like a ghost. There was no sound, no evidence that his feet touched the earth. His breathing never changed, never indicated any effort. But the trees flew by at deadly speeds, always missing us by inches."(TW13) This speed is terrifying to Bella, but merely "second nature"(TW13) to Edward.
We see the speed of all the Cullens again during the baseball game. Emmett is described as "a blur around the bases" (TW17) and Edward can move so quickly that he is able to intercept a ball that has been hit "like a meteor." (TW17) Esme tells us that Edward is the fastest at running, and the baseball game certainly bears this out. Obviously, Edward becomes slower when he is carrying Bella. He is acutely aware that he cannot outrun the visiting vampires with her.
It is not just physical movement that can be exceptionally fast. Speech too can be faster than the human ear can make out. "Edward was growling something too fast for me to understand, but sounded like a string of profanities."(TW18)
His love of speed is also picked up in his driving. As Bella puts it: "He drives like a maniac. It was terrifying." (TW10) He enjoys the speed of the car, and grumbles that 80 is "driving slow." (TW10). He has fast reflexes and Bella notes that it all seemed "effortless" (TW14) to him.