Elana hit all the best points of a post like this: compelling characters need flaws, secrets, fears, emotions, growth, change, and the consequences of stupid mistakes.
Here's what I think, just from what I've learned from the writing process:
Compelling characters need to want something. They have a desire, a drive, a goal. They share how passionate they are about this thing with the reader. The reader gets invested and wants them to achieve said desire. It could be a million bucks, a new friend, or to conquer their fear of spiders. If the character wants it, the reader wants it, too.
Will this goal actually make them happy in the long run? It doesn't matter. They could be right about their goal giving them long-term happiness, and achieve what they set out to do, or they could be mistaken, achieve a moment of self-discovery, and make a change in the game plan mid novel. Both of these options reflect real life. Sometimes we want what we need, and sometimes we want what will end up making us miserable.
Characters need to do something about this desire of theirs. Does this action need to make sense with achieving their goal? It doesn't matter. They can seek their desire directly, and the author can lean on outer obstacles to create tension and challenge. Or the character could hide a secret layer of opposing want beneath their surface want, and sabotage their own goal. Or they could do both. In my opinion, both is the most fun.
What counts in this 'doing things' category is how each perceived success or failure in the context of their goal affects the character. Elation? Depression? Resentment? Frustration? To me, the human condition is nothing more then a desire that fuels a compilation of experimental actions on the universe around us, followed by the resulting impact on our psyche.
What do I look for in a compelling character?