There is only one character who really changes and grows over the course of The Voyage of the Dawn Treader, and that's the unfortunately-named Eustace Clarence Scrubb. Our narrator spends the better part of Chapter 1 making fun of Eustace, from his ridiculous name to his lack of friends to his poor choice of reading material. But the narrator is also careful to stress that Eustace has been corrupted by his progressive parents, who are more interested in jumping on the latest bandwagon fads than they are in raising a son with strong ethical values. Eustace's life is devoid of art, literature, and God, and as a result he is a puny, selfish, unpleasant person. We feel sorry for his cousins, Edmund and Lucy, who have to spend so much time with him.
When Eustace first enters Narnia, he simply doesn't understand what he's seeing. He's never read fantasy novels and, as the narrator explains, has zero imagination: "he was quite incapable of making anything up himself" (1.10). Eustace constantly compares the quasi-medieval world of Narnia to 20th-century England, and England always comes out on top. He thinks of civilization as mechanization and doesn't understand that, to the Narnians, civilization is ethical rather than technological. As Eustace tries to apply the faddish ideas his parents have raised him with to life on the Dawn Treader, hilarity and irritation repeatedly ensue.
It doesn't take long for Eustace to go through a big transformation that makes his exterior form match his internal state. When he discovers a dragon's cave and goes to sleep on its treasure hoard, he wakes up to find that he himself has turned into a dragon – a creature he's never even heard of, since he doesn't read fantasy literature. His character change begins right away: being a dragon makes him realize that he's been a pain in the butt to his shipmates all along, and now it's worse than ever. He starts to want to be liked and to help everyone else. Once this longing is strong enough, he is visited by Aslan, who completes his transformation by helping him shed his dragonish exterior and baptizing him in a magical pool.
After his stint as a dragon, Eustace really begins to change, and by the end of the novel he is nearly as brave and loyal as Caspian and Edmund. However, Aslan hints that Eustace might come back to Narnia someday, which suggests to us that his character development isn't quite complete. If you want to read more about Eustace, check out the next chronicle of Narnia, The Silver Chair.