What makes a compelling character in a book or movie?

Hyperink.com Question of the Month
I like characters that are multidimensional and multifaceted in both books and movies. For instance, my son and I have read every book in the Harry Potter series many times and we still come up with new facets of the characters that we never noticed before.
Compelling characters are those that are neither black nor white but crafted in shades of grey. For instance, was Snape good or bad? In the first few books, we tended to hate Snape with a passion. Harry was more or less the victim and Snape seemed vicious and unreasonable. But as the entire story evolved, Snape seemed more a victim than a persecutor and my sympathies slowly were for Snape. Even then I don't think I switched completely from Harry to Snape. Harry remained my favourite but I did not dislike Snape quite so much. Rather I disliked him but could find it in me to forgive him for the vicious attacks on Harry. When he killed Dumbledore and walked out, I quite detested him but in the last book, it was obvious that he was actually saving Dumbledore from humiliation and pain.
The see-saw of emotions that this revelation of Snape's or any other character in a book or a movie is what makes the character interesting. After all, in real life too nobody's character is good or bad. They are a mixture of reactions to circumstances and intrinsic nature.