Does Edward ever bite Bella? I'm confused about that because she wanted him to so badly in Twilight.

In Twilight, the first book, Edward does not bite Bella except for when he sucks the venom out of her from James' bite. I don't think that's exactly what you mean though. However, if you continue to read the books, a great conflict that occurs through many of them is that Bella wants Edward to bite her, and in doing so turn her into a vampire. Edward is in opposition to this, and argues with Bella repeatedly about turning her into a vampire. Edward does not wish to turn her, whereas Bella wishes to become a vampire. Edward even offers her a deal, that he will be the one to turn her as she wishes, only if she will marry him. Whether Bella takes that deal, and whether she actually becomes a vampire, I'll leave for you to find out as you read on.