A set of words and/or phrases which a given cultural group uses, and which people outside the group may not entirely understand. Usually phrases in the cultural idiom of a given group will have been derived from an experience common to the people in that group. A phrase such as the very annoying "Whassssssuppppp?!!" will have significant connotations to most Americans, for example, but may not be recognized by others except as a greeting. The American culture (in the form of television and advertising) has created a new connotation for this phrase, and thus it has been inducted into the cultural idiom of the USA.
Cultural idiom can be a product of any group of people who have enough characteristics in common that they may be called a culture or subculture. Denizens of a school may associate a phrase with a person at that school, for instance. Cliques may (and generally do) have a set of phrases all their own. If there can be said to be a culture between two people in a relationship, and I think there can, then they may have their own cultural idiom.
While cultural idiom is obviously related to slang, it is not quite the same thing. Slang is generally language changed in some way from formal to informal. This changes how people use a given word, but it does not necessarily change the meaning of the word. A word in a cultural idiom, in contrast, has gained another meaning. For example, in French, barbe is used to mean "boring". However, barbe literally means "beard". In this usage, the meaning of the word has not changed; the implied connotation has. And people outside the French culture probably have no idea what it means to be called a beard.
Also related to the inside joke, but not necessarily funny.