There are three basic kinds of character arcs.
Flat character arcs: In these arcs, the protagonist is great, but the world is not great, and the protagonist’s quest is to fix the world. At the end, the protagonist is still the same, but the world has improved. The Honor Harrington series by David Weber features a flat arc protagonist; Honor Harrington is an exemplary naval officer who sticks to what she knows is right despite everyone being out to get her, and in the end, she prevents tragedy. If you like military science fiction with strong characters, try On Basilisk Station, the first Harrington book.
Positive character arcs: In these arcs, the protagonist starts off in a bad or broken place and ends up in a good or healed place. In these stories, the protagonists’s interactions with the world and with other characters improves him, and that improvement leads him to victory over the villain. Most, if not all, romance novels are positive character arcs, and positive arcs are by far the most common arc in today’s commercial fiction.
Negative character arcs: These are also called tragedies. In them, the character does what he is not supposed to do, and he loses as a result. I can think of two famous modern tragedies offhand: The Godfather and the first Star Wars trilogy (episodes 1-3). A lesser-known tragedy is the Edward James Olmos movie American Me.
For an example of a positive character arc, I’ll use Lois McMaster Bujold’s exemplary book A Civil Campaign, which is a science fiction “comedy of errors” book featuring a strong romantic subplot. (It’s around #10 in the Vorkosiverse series, but can be read as a stand-alone.) There’s an example of a negative character arc after this one.
Miles Vorkosigan, the main character, is very disabled on a planet that hates disability, and as such he has had to BS his way through life. He falls in love with a recent widow, Ekaterin, who wants nothing to do with marriage in the previous book. In this book, they happen to be in the same city, so he decides to woo her. However, her husband died last month and she is decidedly not in the mood. So he decides to ambush her by asking her to build him a garden. She wants to be a professional landscape designer, and he’s an aristocrat with a convenient empty plot of land. She jumps at the chance and goes to work for him.
Miles then goes around and tells everyone he knows that he’s wooing her and that he wants to marry her. His friends all tell him he’s being a dork by ambushing her in this fashion, but he ignores them. He invites her and her elderly relatives to a dinner party he’s throwing with a bunch of other guests, and reluctantly she agrees to go. While at the dinner party (this is the Midpoint), one of his older guests, a man with memory problems, asks him if he’s asked her to marry him yet. He freaks out and, on the spot, proposes to Ekaterin. She roundly rejects him, quits her job as his landscape designer, and decamps on the spot.
Miles becomes very depressed and his best beloved relatives feed him his head on a plate, not unkindly. He figures out what he did wrong and takes several steps to get back into her good graces. He improves himself and his character. Meanwhile, Ekaterin’s in-law from her previous marriage decides to take her son away from her because she’s hanging out with weirdos like Miles. If she sees Miles again, no kid. She tells Miles this. Everyone is dejected. (“Dark Night of the soul.”)
Miles does what Miles does best and mostly outmaneuvers his opponents in a court-like setting, but it’s up to Ekaterin to save the day. She does so. They agree to marry, and the book ends.
There’s a lot more to it than that–there are very funny side-plots, and the context is during the lead-up to the Emperor’s wedding (he’s Miles’s foster-brother), but that’s the arc. Miles starts out by thinking he can ambush the widow into loving him and comes to realize that only stark honesty and vulnerability, not manipulation, will get him the love he’s looking for.
In a negative arc, the protagonist is supposed to be doing the same thing–addressing his short-comings and improving his life, but he consistently makes all of the wrong choices and ends up failing. In Star Wars III, Revenge of the Sith, Anakin Skywalker didn’t have to massacre anyone. He didn’t have to choose fear and power over love and faith. But he picked fear and power, and ended up being a half-barbecued hot dog of a man who turns into the Darth Vader we all love today. (If you have not seen this, it’s the “lightest” of the three tragedies I’ve named here, with American Me being the hardest to watch. American Me starts with a rape and devolves from there; don’t watch it if you’re tender-hearted, and it’s inappropriate for people in their mid-teens or younger. If you want the plot, check Wikipedia. Alternatively, you can watch any Shakespearean tragedy. King Lear is probably the best one for watching a negative arc in action.)