Who is Jaime Lannister from A Storm of Swords and what is their importance?

Jaime Lannister from A Storm of Swords is the son of Tywin Lannister. He has had a lifelong incestuous relationship with his sister, Cersei and they have three children as a result. The realm considers the children to be Robert Baratheon's (Cersei's now deceased husband). Jaime, in this novel, has been released from his captivity by Catelyn Stark and is being taken to Kings Landing by Brienne of Tarth and Sir Cleos. He is to be exchanged for Catelyn's daughters.

Jaime has the nickname of Kingslayer because he betrayed his oath as a Kings guard and killed the very king he had sword to protect. On the way to Kingslanding they are attacked and taken to Roose Bolton who treats them horrible. Jaime has his fighting hand chopped off. They make him wear it around his neck like a jewel. Later, when he is released, he comes back for Brienne, having a momentary bout of conscience. He begins to mature after the loss of his hand and begins to see things from a different perspective. He wants Cersei to come out with their relationship, but she won't.