It starts off from the point in New Moon when Edward leaves Bella and she finds Jacob but she is still hurting inside and still reminiscing on the times she spent with Edward and still doesn't let go of him. She thinks about him all the time even when she's with Jacob. The black and white scenes are the memories of her and Edward. She spends time with Jacob and they get closer but then she cliff dives and all the things happen with the Volturi etc. But after, when Bella gets Edward back, she reminisces on the memories she had with Jacob hence, the clips of Jacob are in black and white.
The song is so amazing it makes me cry. To me, it's about how Bella still wants Edward no matter what. Even if the world ends, she still wants him. And even if he leaves and never wants to see her again, she still wants him back.