Well-drawn characters should feel like real people - with strong personalities, quirks, weight issues, Bachelor's degrees, huge DVD collections, sibling rivalries, sleep-walking issues, etc. They should be full and deep and intricate and complex, just like real people.
Which leads me to state something slightly controversial: I believe that real people don't change. Now, certainly, people can become jaded or more open-minded, can develop new musical tastes, can even change political parties throughout their lifetimes. But the core of who they are, their very essence, generally does not change: core values/beliefs, personality traits, basic view of the world. These things, I've found, usually remain the same, even over decades of time.
Likewise, characters in our books/stories/plays should not drastically change. Of course, they should evolve, grow, learn from mistakes, or else why write about them in the first place? Nobody lives in a stagnant place. There's always room for growth and improvement and lessons learned. A character should grow from Point A to Point Z, surely.
But - the very core of who the character is should remain the same. Because when it doesn't, then it leads the character to do something...well, out-of-character. And that frustrates readers. They will pause the reading, scratch their heads and furrow their brows, and say, "That character would never do/say something like that." Even worse, readers might stop trusting the writer altogether, wondering if the writer knows his/her characters at all.
I'll let you in on a little secret. One of my guilty pleasures is watching The Young and the Restless. (*waits for tomatoes to be thrown at the screen* LOL) I've watched it since I was in college, so I know these characters pretty darn well. One time, a few years back, Victor Newman did something TOTALLY out-of-character. I don't even remember what it was, but it had me practically talking to the screen: "What?! Victor would never do that! How stupid!" I can't tell you how frustrated I was that the writers had gotten so lazy. Or sloppy. Or careless. Whatever the case, as the viewer, I was pointing my finger directly at them.
As a writer, trust that your readers are smart, that they will care for and know your characters -- well enough to hold you accountable for sloppy/lazy/careless writing.
So, how do we avoid angering our readers, and keeping our characters "in character?" Know the characters. Well. Better than anyone else. Know their habits, their morning routine, how many sugars go into their coffee. Know their dating history, their future goals, how much they love or hate their current career. Have a conversation with them (internally, unless you don't mind that your friends/family might think you're going a little nuts). Know the way they talk, walk, whisper, dream, hold grudges, love, and hate. Know everything about them.
Then, the likelihood of them doing anything "out-of-character" will diminish drastically. And you will have happy readers. Which is always a good thing. ;-)