Propriety is following what is socially acceptable in speech and behavior. Your little brother might offend your sense of propriety by burping loudly at the dinner table.
Propriety is like the noun form of "proper" and "appropriate." It comes from the same root as the word property, in the sense of "a distinctive character or quality." The opposite of propriety is impropriety, meaning "rude or inappropriate." If you're someone who is overly concerned with propriety, your friends beg you to loosen up now and then.