A/N Okay, thanks to everyone whose reviewed ( 3 so far) I understand it's a bit confusing, so basically, it's like this.
The Volturi have this vampire who can manipulate people and time. He's like a super powerful version of Jasper, only ... well, evil. He pulled on Esme's emotion's and made it really hard to concentrate on anything except wanting a child - and he never actually told her she'd have to go back to Charles to have one. So then the vampire -whose name is Halik, by the way - changed time so that Esme never meet Carlisle, and never ran away from Charles .
The point of the Volturi doing this was to try and weaken the Cullen's as a family unit; without Esme there their not so close to each other, so there's not so much of a "threat" to the Volturi.
The mystery vampire (Rachel ) is someone Carlisle knew and grew really close to when they were both new-born's. They went to the Volturi and Aro wanted her to stay, because she can control darkness. But she wanted to leave with Carlisle (they were really good friends , and she was a vegitarian as well.) Aro got Halik to change the timelines so he found her before Carlisle and she became part of her family.
The wolves part in chapter three is supposed to be a flashback of the part in Breaking Dawn Part One, where Carlisle , Esme and another Cullen had to go and hunt , even though Sam's pack were trying to kill Nessie . Esme gets attacked by one of the wolves and Carlisle snarls at them as he pulls her away. (I think that happened in the film, or did I just make it up ?) . Basically, the Rachel's snarling to make Charles stop attacking Esme reminded Esme of when Carlisle warned the wolves to back off in breaking dawn.
Hope this clears it up a bit, thanks so much for reading!