An arc is the journey a character makes that reveals how he's grown by story's end. Story is about change, and the events in a story reveal that change, usually through the characters. Without change, there's no story.
Luke was an obnoxious farm boy at the start of his arc; and he was a powerful Jedi at-peace-with-himself by the end of it. Everything that happened to him in between is the arc.
Example: Wizard of Oz. Dorothy starts out as an unhappy girl who wants adventure and runs away from home to find it. The tornado takes her away from home and sets her on an adventure through the land of Oz where she's constantly trying to find her way back. She came to learn there's no place like home. The adventure in Oz was Dorothy's character arc.
I heard somewhere that the character arc is what changes the character goes through during the course of the story, which is basically what everyone else said. Sounds like you "got it" anyway.
Like a plot arc, a character arc often has a climax near the end of the story (usually a bit before the plot climax). Throughout the arc the character is under pressure to change; at the climax they either do change, or strengthen their resolve to never change (in that particular way).
Some characters don't change at all -- by the end of the story they're more or less the same as when they started. James Bond is the classic example. He may be affected temporarily by events that take place in the novel or the film but at the end he's still James Bond.
But a visible change to the character -- showing how they become a different person during the course of the story -- can be much more satisfying.
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