"a brief saying or action making a point, attributed to some specified person or something corresponding to a person." (Theon 96, Kennedy)The chreia is usually the third exercise in the ancient progymnasmata.
A chreia is thus a very broad category of either sayings or actions. What distinguishes it from a maxim is that a maxim can be anonymous, and only in a chreia can there be actions as well as sayings.
Much work has been done on the chreia in New Testament studies, so I will try not to rehash that information here (see Hock, Ronald F. and Edward N. O'neil. The Chreia and Ancient Rhetoric: Classroom Exercises. Leiden: Brill, 2002.).
The chreia, having such a broad definition, makes it useful for evaluating much of the work in the New Testament. Almost any brief saying or action done by Jesus in the gospels could be considered a chreia and evaluated according to the discussions of the preliminary exercise.
One interesting example might be the story of the woman caught in adultery in John chapter 8. Theon says that chreias can be verbal, actional, or mixed. That is, a chreia can be a saying, or an action, or may contain both. The story in John 8 would be a mixed chreia containing both saying and action. Twice in the story Jesus bends down and write in the dirt, hence the action part of the chreia. But, he also adds a saying, "Let anyone among you who is without sin be the first to throw a stone at her," hence the verbal part of the chreia.
Some of the interesting aspects of chreias are that they can be inflected in different grammatical cases, expanded or contracted, confirmed or refuted, and restated. Of specific interest to NT scholars might be the idea of expansion or contraction of sayings. For example, in source critical discussions it is often assumed that the shorter version of a saying is the more original. Yet, with the knowledge of the progymnasmata, it becomes clear that students learned both to expand and contract source material, erasing the certainty of the more original form.