A character arc is the development that takes place in a character's emotional life over the course of the story. They start emotionally at A and end up at B. With luck, they've also been through C, D, E, F, G etc on their way to B, but put simply, by the end they have changed. Usually they have learned something about themselves and/or the way the world works.
Take The Great Gatsby. Nick Carraway, the narrator, comes East. He's from the Mid-West, and thinks his cousin Daisy is wonderful with her rich husband and glamorous Eastern lifestyle. Mid-Western values of honesty, hard work, thrift etc are forgotten as he gets sucked into a more sophisticated way of life. But by the end he has changed. He returns home, disillusioned by the East, emotionally scarred and generally sadder and wiser than when he started. That's his character arc.
Isabel in Adultery for Beginners starts out as being entirely dependent on her husband, not just financially but also emotionally. By the end, she is reaching towards financial independence, she is taking responsibility for her own life and actions, she is making her own future instead on relying on others. She has learned about self-reliance and self-determination, and when she establishes a new relationship it will be as equals.
Often, when the novel starts we see our main character as having this virtue, that fault. Over the course of the story we usually learn why they have those virtues/faults. We learn what are the problems created by those virtues/faults. We learn how they're going to overcome those specific problems and crucially, how THEY are learning not to make those mistakes again by recognising their virtues/faults and changing.
In Adultery for Beginners, my editor said she wanted only a hint that Isabel and Adam were going to get together, rather than a full blown passionate embrace, explaining that readers could fill in the gaps for themselves. She didn't explain further than that - she may not have known why it was deep down the right thing to do. Neither did I; I just accepted the situation and amended it.
In retrospect I know why it was right for Isabel. Isabel at the beginning of the novel is impetuous; she behaves like a child wanting things now. She rushes headlong into a relationship, and gets badly burned. But by the end she has grown up. She has learned not to do that again. She will take things with Adam slowly, and let the relationship develop at its own pace.
I believe that one of the reasons we read fiction is to discover how other people deal with change. Without change there is no point to reading. In a novel the character will change on many levels, in a short story there will usually be space for only one or two changes. But change there will be.