A "symbol", in a literary sense, is an object in the story that represents something else, such as a dove representing peace, or a child's doll representing innocence.
The two most glaring symbols I can think of in No Country For Old Men are both carried by the antagonist, Anton Chigurh: the coin, and the bolt gun he uses to kill his victims and enter buildings.
The coin he uses to determine whether a potential victim will live or die, believing he surrenders his will as a killer to the will of the coin. It is a symbol of random chance, in my opinion, that a coin could find its way to Chigurh and then to his victims by pure random paths, their deaths reduced to a coin flip, as random as if they were hit by a bus or killed in war.
The bolt gun, actually called a Captive Bolt Gun, used to kill cattle, is instead used by Chigurh to murder humans. I believe McCarthy intended it to be a symbol of slaughter, in this case not of cows, but of men, and in the same manner as cattle - dehumanized, their value reduced to that of mere animals.