The Sheriff's dream of his father is symbolism for the old generation not leaving a pure, clear path for the new generation. The dream was something to the effect of seeing his father on horseback at night, carrying a torch... and his father simply riding by him, off into the darkness.
The Sheriff related to his father, in this dream, because he, too, was becoming a relic of the past, in his retirement -- the old generation leaving the keys to the castle to the new generation.
If you noticed, the Sheriff did a poor job of teaching the young deputy his ways, and cuing him in on his suspicions -- the young deputy, who will take his place in retirement.
The Sheriff was in stoic awe of the way things were dissolving in the world, and wanted to help, but really couldn't... It's a downward spiral, hence "No Country for Old Men".
Moss represented the new generation. Moss wasn't a bad man, in essence... but he was confused with what was right and wrong. This was represented in his action, but especially when he returned to the proverbial "scene of the crime" to give the suffering Mexican some water, which he should have done hours ago. That showed that, while he makes mistakes, there is a part of him that is good... but the old generation didn't teach him well, and his selfish mistakes killed him and his wife.
Chigurh was an element outside of the new and old generations. I think this is so because of the Sheriff's voice over at the beginning of the film, about the sociopath who shot that judge in cold blood, generations ago.
Chigurh made off with the money and killed Moss' wife. I know this because there were clues sprinkled throughout the film.
When Chigurh left her home, he checked his boots for blood. Throughout the film, Chigurh avoided getting blood on his clothes. When he strangled the police officer with his handcuffs, and blood spurted from the cop's neck, Chigurh turned his head to avoid and blood to splash on his face. When Chigurh killed the Mexicans in that hotel room, he shielded himself with the shower curtain when he shot the Mexican hiding in the bath tub. When Chigurh killed Carson Wells, and when the blood pool began flowing towards his boots, he put his feet up on the table. (I think there was one more instance, but I can't remember right now.)
He was a creature of habit and had his own, twisted principles. The "If you see him, you're as good as dead", or whatever the line was that Carson Wells said about him, is a part of his principles -- along with keeping his clothes fresh and clean.
There were two instances in the film where he didn't kill people who saw him. One was the fat broad in the trailer park. You could tell he was going to, but didn't... because a toilet flushed in the background - someone else was there - and he was too much of a pro to make a mistake.
The other instance was after his car accident -- where he bypassed his own principles by telling the two boys, "You didn't see me." That was more for himself, then for what he wanted them to tell the cops.
The other part of his principles was the coin, and this is the other clue how I know he killed Moss' wife (aside from him checking his boots) -- she refused to call it. He promised Moss that he'd kill her, and he did -- but she refused to call it. He broke his own law. He was a force; a phantom... He'd always know where you were, he'd always find the money, he'd always be able to walk among us without losing a step... but because he broke his own law, that's why he got nailed in that car accident. It knocked the stars out of alignment, and you could feel that he knew something bad was going to happen while he was driving, and gazing into the rearview mirror.
(The chance of getting in a car accident also eloquently parallels that of a coin toss.)
And I believe that he survived that accident. A compound fracture is lethal, because of infection... but because he was a such a force, and because they showed how he handled his shotgun wound, he definitely survived.
Evil never dies.