Bell is a rural Texas county Sheriff who has spent his career bringing a sense of order to his corner of the world. Like many in law enforcement, the world is black and white, and while crime can't be condoned, the motivations for it and the people behind it can, most times, at least be understood.
We get an early clue as to what's eating Sheriff Bell in the first chapter, when he talks about a killer he arrested and testified against being executed. The vicious nature of the murderer, and the utter lack of remorse he shows for his crimes bother Bell's sense of order. He can't explain it, and it gnaws at him.
The same is true of Chigurh. He is the modern drug gang assassin, while Bell can only be the small town Sheriff. No matter how good a cop he is, he cannot come to understand Chigurh or others like him. It makes him question his faith in God, and the belief system he has nourished all throughout his life. He wants to quit, in part because of the evil he's facing in Chigurh, and in the end he does, but not without feeling that somehow, he has surrendered.