A character arc is the change a character goes through as a result of what he experiences from the start to the finish of the story.
Not every story has or needs its protagonist or another character to arc. The very first James Bond films come to mind.
But if the story does have one, it’s usually the protagonist who arcs. And usually the end of the protagonist’s arc “proves” the theme of the piece.
HISTORY OF VIOLENCE explores a man caught between his violent past and his pacifist present…making us question whether the use of violence to solve a problem is always wrong.
When the protagonist starts out, he’s against the use of any violence for any reason. But when the violence of his past is revealed and it threatens the lives of those he loves in the present, and turning the other cheek fails, he chooses to use violence to solve the problem. And it does solve the problem. But it forever changes his present and the cost seems to be a dear one to not only him, but to those he was trying to save in the first place, his family.
In plots where the protagonist doesn’t arc, it can happen that the protagonist by his actions and beliefs, causes another or other characters to arc. It’s rare, but it does happen.
In FERRIS BUELLER’S DAY OFF the theme is: Sometimes you just gotta say WTF and enjoy life.
Ferris, the protagonist, is no doubt the person who patented this attitude. He’s the poster boy for Carpe Diem.
It’s his uptight, angst-ridden, morose best friend and his annoyingly goody-two-shoes sister who don’t even know that this devil-may-care approach to life exists. So Ferris spends his time trying to get his best friend to see the light and eventually succeeds, which also changes his sister.
The point of all this is that before beginning the screenwriting process it can be helpful to clarify the protagonist’s (or other character’s) character arc(s) and subsequently identify the theme as well.