A dynamic character, sometimes also called a round character, is one who experiences or demonstrates significant change or personal transformation during the course of a story. Dynamic characters are often contrasted with static or flat characters, who do not experience or demonstrate any real modification in thought or action between the beginning and end of a narrative. Most protagonists are dynamic characters, and this is in part why we are able to invest in them as readers.
In F. Scott Fitzgerald's "The Great Gatsby," the narrator, Nick Carraway, leaves the Midwest for New York to start his career and to explore various possibilities. He is very innocent and is in many ways naive about people and the ways of the world. Nick sees Jay Gatsby at first as a sincere, self-made embodiment of the success that every person hopes for. Throughout the novel, however, Nick becomes disillusioned as he learns of Gatsby's less forthright dealings and sees his desperation. Nick himself also changes from being an innocent and inexperienced young man to one who has witnessed crimes, excess, cruelty and sadness, and he is even cruel (unintentionally) to a woman who liked and trusted him. Because of the deepening and transformation of his perspective and person, Nick is a good example of a dynamic character.
Another example of a dynamic character is Elizabeth Bennet in Jane Austen's novel "Pride and Prejudice". At the beginning of the story Elizabeth detests Mr. Darcy and sees only his faults. She is stubborn and unyielding in this way. However, as the book progresses, she learns of his good qualities and noble actions, and her opinion of him changes. She also, through making mistakes in judgment, gains greater knowledge of herself and her own faults and becomes more humble.
In Virginia Woolf's stream-of-consciousness novel "Mrs. Dalloway," Clarissa Dalloway is also a dynamic character. Although she does not alter her personality obviously or change the way she does things, she has a greater awareness and acceptance of the world and her temporal and social place within it at the end of the novel than she does at the beginning. Over the course of one day, she rehashes the events of her lifetime, and she seems to reach a greater quietude and connection to the present moment as the story ends. Although her "dynamism" is more subtle than dramatic, her character is complex and experiences minor yet significant transformations.