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At the beginning of the story, Little Red Riding Hood is a sweet, innocent thing. But we know that by the third act she'll be tough or she'll be toast. Her evolution is called a character arc. It's one of the most important elements of storytelling, and for a simple reason: when a character grows, we experience her change vicariously and are transformed along with her.
We get involved with a character because we want to know if his dilemma will change him. Will it force him to overcome a deep flaw (Liar, Liar), realize his potential (Rocky), or heal an emotional wound (Silence of the Lambs)? We want to see the hero grow and change, because it gives us hope for ourselves.
The character needs to be capable of change from the very beginning, otherwise the change won't ring true. Take Rocky, written by Sylvester Stallone. Rocky Balboa is an underdog. The odds are stacked against him, but he's determined. It's this quality that will help him transform from a nobody to someone who has achieved his potential. The change has to happen gradually. If it's too sudden, it will seem forced and implausible.
Another way to think of the character arc is as a map of your character's beginning, middle, and end. Your character might start off as selfish, like Jack Nicholson in As Good as it Gets (written by Mark Andrus and James L. Brooks). As the story progresses, he faces situations and conflicts that increase his self-awareness. By the end of the movie, the character has let go of his original identity and has become more generous and thoughtful.
Make sure that at least your protagonist has an arc. Depending on your story, the antagonist and some of the supporting characters might have arcs too. A character arc doesn't necessarily require improvement. If you're going for a down ending, your protagonist will change for the worse (Chinatown, Raging Bull).