Villain Character Arc
First, a quick refresher of the three main definitions I'm working with.
Protag/Hero: The one with the most to lose -- the main character (Frodo, Will Turner)
Antag.: Oppose the Hero -- supporting characters to the villain (Gollum, Norrington)
Villain: The one with the most to gain -- the main "bad guy" (Sauron, Davy Jones)
A little sidenote: one way to clarify between villain and antagonist. An antagonist can, and often will, switch sides. Norrington was an antagonist, then became a good guy. Gollum briefly changes to Frodo's side.
Villains never switch truly. (Sauron, Davy Jones, Palpatine)
Hopefully, by now you have already or are planning to read at least a few writing books on the craft. If not, I won't shun you, but it's a very good idea to do so. Especially written by the same authors that you like to read--since you likely share some of their preferences and stylings.
If you have read, then this is a good refresher. Or you can just hop on down to near the bottom.
Character arcs are simply, the pathway of change.
That is, your Hero absolutely must start at point A in his personality/knowledge/etc. and slowly be changed by the progress of your story and events around him, so that when he reaches point B (or C, or D) he is no longer the same Hero.
sidenote: If there is no character change...there is no story. 'Nuff said.
The most common format is a three-part arc, though you can have as many parts as you like. Most of humanity seems hard-wired to find a sort of symmetry in three pieces though. (Three Little Pigs, Three Bears, Three Wishes....Four Pigs just doesn't have the same ring, does it?)
In your Hero's character arc, he should begin at the point of change, when some external or internal event (though typically the former) prompts him to change his locale, his mood, his action, his emotion, etc. This is the baseline of your arc.
Examples: When Frodo receives the ring, when Elizabeth is kidnapped, when Luke cleans C-3PO...
A arc then as rising or mounting action, where the change prompts another change, and another. In the process there should always be at least one, if not multiple, problems. (3 is usually a good number...) And each problem is harder to defeat than the next.
Examples: Crossing the Dead Marshes/Black Gate is Closed, Jack Sparrow's trickery, Imperial troops...
The final problem is the climax. Directly before this final problem (which your Hero should find a way to defeat) should fall your lowest portion of your arc, when all seems lost. Everything should come crashing down around your Hero.
Examples: Frodo stung by Shelob/collapsed on Mt. Doom, Will captured by Bubossa in the cavern, trapped with Palpatine and Vader...
The Hero should then rise above it (finale/climax/resolution) and in so doing produce the final change. From there, either the story ends or there's a gradual tapering off that wraps things up.
Examples: Finale--ring tossed into lava, tapering into the return to the Shire/Grey Havens, Finale--freeing Bubossa from the curse, tapering into Will and Elizabeth running off together,
Finale--blowing up Death Star, tapering into the "medal ceremony" for Hans and Chewbacca.
But how does this apply to villains, you ask.
Just as the Hero has a character arc, a path of change. Did you know that villains (and to a lesser extent) antagonists alike have one? I don't mean the wishy-washy bad guy transforms into good arc either. Sauron has a character arc, and Bubossa, and Darth Vader and Palpatine.
There is one important thing to remember, however. Opposites.
Your Hero and Villain character arcs should interact and influence one another, but in utterly opposite ways. Every time your Hero succeeds, your Villain fails. Every time Hero fails, Villain succeeds.
I'm a visual sort of person, so, for your viewing pleasure: This is a simplistic graphic of two character arcs--the Hero and the Villain. There's blue for Hero and red for Villain, and when those mix, you get purple.