If something is proverbial, it's referred to in a familiar saying. If your little brother knocks over his milk and starts crying, you might think of the proverbial spilled milk.
Proverb is the root of proverbial, and it comes from the Latin word proverbium, “a common saying.” Proverbs are little stories or expressions that usually teach a lesson, like "Don't cry over spilled milk," which means "It's a waste of time to be upset about something that can't be helped." You could say to your dog, "Well, aren't you the proverbial best friend?" or tell your sister, who's dyed her hair purple, "You stick out like the proverbial sore thumb."