Candy's old sheepdog is a very important part of the story. He is symbolic of Candy, and Lennie for that matter. That is, something that "ain't no good to hisself no more", something to be put out of its misery.
So in the bunkhouse after work one evening, Carlson convinces Candy to let him take the sheepdog up the hill and shoot it. Slim says he can have one of his pups as a replacement, but it's very difficult for Candy to give up the dog. Right there in front of the other field hands, he feels he has to, but instantly regrets it.
Later in the story, Candy says how he wishes that, when he ain't no good no more, someone would shoot him. Instead, he says, they'll "can" him, and he won't have any place to go.
It seems like a minor part of the story at first, but actually it's a very important one to Steinbeck's overall message.