If you are referring to her demand that he be the one to turn her into a vampire, he said he wanted her to marry him first, as a human. Carlisle had already promised Bella that he would change her into a vampire after she graduated from high school; but then Bella shared with Edward that she would prefer it if Edward were the one to change her. Once Edward learns this, he has leverage over Bella, and he tries to get her to postpone the change as long as possible. Bella doesn't want to reach the human age of 20 before she turns; if Edward is a teenager forever, she wants to be one as well. But when Edward's condition is marriage, she freaks out on him. She doesn't want to get married right out of high school; she worries about the judgment she would receive from her mother and other people who would think it was ridiculous for her to get married right after high school. Edward is okay with postponing the wedding, because it will mean postponing her change into a vampire. But he doesn't understand her rationality, since he is from a time when people married at a younger age.
Her conditions was that she would move in with Edward since she was not a minor and Charlie was not so happy about this. Edward said that she would marry him before he turned her into a vampire and suffer eternal damnation.
In New Moon the only ultimatums she gave out was to Charlie and that was if he didn't let Edward come around/ let her date him she was going to move out and live with him especially since she wasn't a minor now. That really made Charlie made but he accepted neither the less. However Edward doesn't give Bella condition until Eclipse and that would be that she marry's him before she chooses "eternal damnation" as he would like to put it haha.
his condition was she had to marry him first