The Golden Compass
Iorek Byrnison is one of the panserbjørne – a race of armored polar bears that live in the land of Svalbard. Once a prince, he has been banished from the kingdom for killing another bear in the heat of passion. We eventually find out that Mrs. Coulter is responsible for his banishment. Iorek's arch nemesis is the bad bear king, Iofur Raknison. He is longtime friends with Lee Scoresby, whom he knew before the events of The Golden Compass take place. Iorek and Lyra become close friends over the course of the novel.
Iorek's story is a happy, redemptive one – which is a good thing, because a lot of depressing stuff goes down in The Golden Compass. He goes from being an outcast to reclaiming his place on the throne in Svalbard. When we first meet him he is being held hostage by humans in Trollesund (Chapter 10). He is depressed and guzzling liquor in an attempt to forget his sorrows. Though he seems pretty broken down, Iorek regains his strength when Lyra helps him find his armor and frees him from captivity. At this point Iorek joins the travelers headed to the North (Chapter 11). After saving the children at Bolvangar, Iorek (with Lyra's help) fights the bad bear king Iofur and reclaims his rightful place.
Iorek's redemption, we should note, is also the redemption of the panserbjørne as a whole. Iofur had the bears acting like humans, which weakened them. Iorek will ensure that the bears continue along their right path as bears – he represents the bears' destiny:
But Iorek and Iofur were more than just two bears. There were two kinds of beardom opposed here, two futures, two destinies. Iofur had begun to take them in one direction, and Iorek would take them in another, and in the same moment, one future would close forever as the other began to unfold. (20.35)
While most characters in this novel have an animal soul to keep them company, bears don't have daemons; instead Iorek's armor is his soul. As Iorek says:
"My armor is made of sky iron, made for me. A bear's armor is his soul, just as your daemon is your soul. You might as well take him away" – indicating Pantalaimon – "and replace him with a doll full of sawdust. That is the difference. Now where is my armor?" (11.136)
Armor made of sky iron is part of Iorek's core identity; without it, he feels incomplete.
Iorek and Lyra
Lyra and Iorek's journeys are deeply intertwined – their fates are connected. Lyra helps restore Iorek to the throne in Svalbard, and Iorek accompanies Lyra to find her father's laboratory. Lyra often compares Iorek to Lord Asriel (13.93), and he becomes a kind of replacement father figure for her. As she tells Lord Asriel:
"I love Farder Coram, and I love Iorek Byrnison; I love an armored bear more'n I love my father. And I bet Iorek Byrnison loves me more'n you do." (21.75)