The problem with character arc
Character arc refers to the path your hero (and supporting actors) follow throughout your story. In essence, your hero must overcome his or her personal, financial, romantic, or spiritual issues in order to save the day and provide the world with a satisfying ending.
It's referred to as the character ARC because no one ever follows a straight line in their learning. There are up and downs in everyone's life and it's just that much more gratifying to watch your characters suffer small setbacks and rewarding victories on their path to the final act where they use everything they have learned throughout the course of your screenplay to vanquish the antagonist (or the problem, or mother nature, etc.).
If all of your characters don't follow some kind of arc, they seem flat and uninteresting. That is why it is imperative that each character (not just your hero) has an issue they must overcome or quirks that make them unique. There is a great ebook about adding dimension to your characters called 1001 Character Quirks for Writing Fiction by Tim King that acts like a primer for getting started. The great thing about this books is that it doesn't tell you what to do. It gives you simple ideas that you can weave into your character's life to make them compelling and it gives them great challenges they can over come.
If you can get past Tim's tragic haircut and cheesy moustache, you find this an invaluable book on getting those ideas flowing.
Quirks and back-story in character arc?
To put it plainly, there has to be well defined reasons for characters to do the things they do. If your hero has a fear of flying, you don't necessarily have to spell out the plane crash that killed their parents, but we as an audience appreciate at least a glimpse of why they won't get on the plane. It makes us empathize with what they are going through.
How much more rich is the Indiana Jones character now that he has a debilitating fear of snakes? I think the movie would still be outstanding without his "issue," but these tiny little nuances make him human instead of the swashbuckling college professor who is completely invincible. Quirks and back-story feeds perfectly into character arc because it grounds an otherwise unbelievable hero.
How does your hero feel about the world?
We all have seen the ubiquitous "crotchety old man" character. He hates change, is most likely a racist, hates kids, and generally demands things always go his way. The world develops around him and he sits there like an island of frustration and thinly veiled hatred.
Does this guy show up in your story?
Bad move! Congratulations, you just put a cliche in your screenplay that essentially got it tossed into the garbage. The world doesn't need any more quintessentially angry old people. Of course, just about every family in the world has someone like this messing up family holidays, but why?
What if you actually give an old guy a reason to be such a jerk? What if he's such a racist because he was a prisoner of war? Or maybe he's afraid of change because he was an orphan and his adopting family disintegrated when his foster mother lost her faith?
It's one thing to be deliciously vicious as an antagonist, but without a great reason, your characters are only cardboard cut-outs. Anyone in your screenplay who does something great (or evil) should have the proper motivation to do so.
Nothing ever works out the way we planned
If your story is a linear "and then this happened... and then this happened..." you will lose your audience before you even get the chance to get them excited about your plot and premise. Without a character arc, the whole point of your story becomes uninteresting and lifeless. This concept is best handled in Rob Parnell's Easy Way to Write system. He covers a bunch of genres and best of all, he teaches how to make your characters irresistible!
Your heroes must be dragged kicking and screaming into the final act. Imagine Toy Story if Buzz Lightyear never realizes he's a child's toy. Real heroes have real problems. There can't be a Superman without a whole bunch of Kryptonite.
Throw everything you can at your hero. After the house burns down and her husband leaves her and her mother gets kidnapped by terrorists, her dog dies. All the while she must be brought to the brink of insanity. Her very essence must be challenged. All the things she believed about what she was or wasn't capable of must be dragged out into the back yard, beaten with an old baseball bat, hosed off, beaten down again, and then propped up in the front yard for the neighbors to ridicule.
In short, nothing should ever be easy for your hero! There's nothing more satisfying than watching someone who's spent their entire lives terrified of heights finally jumping off the high dive to save the burning kittens. As jaded and downright creepy this world can be, we need to have hope that we can overcome our issues in order to save the day. This is character arc! Your hero mirrors our own desire to triumph over the things that continue to limit our abilities to get ahead in life.
Character arc and the things we all know
We all know that smoking is bad and fast food is an artery clogging menace to our quality of life. This is why most heroes start the screenplay from a position of weakness. In Legally Blonde, Elle is just an oblivious girl on the "score a man to support her" track. When thrust into the real world, she conquers it by using her own unique set of skills and very particular point of view to make the lives of everyone around her better and helps her save the day.
Elle knows that there is very little she can do in reality, but with her implacable optimism and sunny nature, she becomes a force to recon with.
There are things your character already knows to help them on their journey. More importantly, there are things they are about to learn about themselves and the ways of the world that allows them to defeat whatever nastiness you have saved up for them at the end. Imagine character arc as all the things you need to know to pass the test at the end. Each mini-obstacle gives the hero either the specialized knowledge or the self-reliance they will need to get them to the final hoorah.
What if there is no final hoorah? Maybe your story ends in tragedy. Perfect! The world needs more sad endings (they might not sell very well in Hollywood, but...) If you have a tragic ending, then your characters still must follow their character arc to their doom. What makes a tragedy that much more tragic? How about your hero finally discovers how magical life is only to still die of cancer in the end.
We don't always need the day to be saved, but what we do need is for your point to resonate and be meaningful. If your characters grow and learn and apply that knowledge, then you have done your job. You've empowered the viewer to rethink their own lives. Your character arc might just change the world...