At the end of the book (and the movie as well) the vampire bites Bella, meaning to kill her and lure the Cullens into fighting him. However, Edward arrives in time to prevent James from killing Bella; instead he just manages to bite her hand. According to the book, when a vampire bites but does not drink all of a person's blood, then the venom from their bite spreads throughout the victim's body and turns them into a vampire. This is a painful process, and it burns like fire through the person's veins. When James bites her, this process begins to happen to Bella. In order to make it stop, Edward must suck the venom from James' bite out, and yet stop in time before he drinks too much of Bella's blood which would kill her. Somehow Edward finds the strength to stop, and Bella survives and remains human.
James bit her to make her into a vampire as well as trying to cause Edward a lot of pain. He was shoving it in their face that he has won by hurting Bella. Every single vampire bite burns since they are shifting into a vampire but Edward drinks and spits the venom out of her to make sure she stays human.
James bit Bella to feed and enjoy her as a reward to the chase and hunt he was putting the Cullen's through which he was gloating that he had won. So yeah her reaction was probably icing on the cake for him. All vampire bites are going to burn because it signifies the change to becoming a vampire however many victims do not live long enough to complete the change. This is the reason Edward sucks the venom out of her so she doesn't change into a vampire.
His name is James, and he bites her because he wants a fight from Edward. That is all part of James's little game, and he wanted to make it harder and funner for himself.