I always seem to gravitate towards characters with a redemption arc. Jaime's got a whole lot of baggage to leave behind and he's trying to be a better person, get his honor back, but without overtly saying it. He's very real. Very complicated.
He's a romantic idealist at heart, who has lost his way.
I much prefer him in the books than in the show, though.
How about you? Any favourite thrones/ASOIAF character?