Chiastic structure, or ring structure to give its less formal name, is the prose equivalent of a Petrarchan Sonnet. Which, for those of you less interested in poetic form, is, I grant you, not much of a clarification. Okay, let’s start again.
Rhyming Scheme
The first half (the octave) of a Petrarchan Sonnet has a very simple rhyming scheme: A B B A. Example, courtesy of Edna St. Vincent Millay:

Chiastic structure is often drawn like this
What lips my lips have kissed, and where, and why,
I have forgotten, and what arms have lain
Under my head till morning; but the rain
Is full of ghosts tonight, that tap and sigh
Now, obviously you can’t make your story structure rhyme. In prose, this is about ideas, scene themes (or, if you want to really play it down to the granular level, actual sentences) being repeated in reverse order to bring the audience back to the same point as they started from but with hopefully a new angle or appreciation for that starting point. Think of the heroic journey – that is, in itself, a basic example of chiastic structure. From a plot structure perspective, one of the simplest examples is Milton’s Paradise Lost:
A: Satan’s sinful actions
B: Entry into Paradise
C: War in heaven (destruction)
C: Creation of the world
B: Loss of paradise
A: Humankind’s sinful actions
So, sin – paradise – destruction/creation – paradise – sin. Which, actually, is a point to note: you can go way beyond A B B A. Milton has A B C; the story of Noah’s Ark has A – J:
A: Noah and his sons (Gen 6:10)
B: All life on earth (6:13:a)
C: Curse on earth (6:13:b)
D: Flood announced (6:7)
E: Ark (6:14-16)
F: All living creatures (6:17–20 )
G: Food (6:21)
H: Animals in man’s hands (7:2–3)
I: Entering the Ark (7:13–16)
J: Waters increase (7:17–20)
J: Waters decrease (8:13–14)
I: Exiting the Ark (8:15–19)
H: Animals (9:2,3)
G: Food (9:3,4)
F: All living creatures (9:10a)
E: Ark (9:10b)
D: No flood in future (9:11)
C: Blessing on earth (9:12–17)
B: All life on earth (9:16)
A: Noah and his sons (9:18,19a)
Purpose in Prose
Chiastic structure has its origins in oral poetry, with a dual purpose of 1) reminding the poet where he was supposed to be getting back to (no, seriously) and 2) making parts of the poem self-contained so they could be recited as stand-alone sections. Both the Iliad and the Odyssey are riddled with chiastic structured sections, as is the Old Testament, the Qu’ran, the Torah, and pieces like Beowulf.
In more modern literature, it’s much more about fulfilling audience expectations. We like symmetry, whether we consciously notice its presence or not. The symmetry of chiastic structure therefore provides a very satisfying sense of closure to a story. The reader remembers this scene or idea from the opening and therefore recognises it as an acceptable ending. Like the hero, the reader has travelled on a circular journey and arrived home with a new understanding of that same old idea. This links very strongly into the causal chains stuff that I talked about last year.
JK Rowling deliberately used chiastic structure in the Harry Potter series. (WARNING: Spoilers ahead. But it’s been, y’know, a while so I figure I’m allowed.) Lupin and Tonks die to mirror the death of James and Lily Potter, as casualties and consequences of war. At the beginning, Hagrid carries Harry away from where Voldemort killed his parents and delivers him to his new home at Privet Drive; at the end, Hagrid carries Harry from the Forbidden Forest where Voldemort ‘killed’ him and delivers him to his friends at Hogwarts.
One potential pitfall (I’m all about the p-based alliteration today, apparently) to be aware of is balancing the demands of chiastic structure against the actual story. If your characters want to do something that breaks the ring cycle, do you let them? Or do you stick to the plan? Rowling is on record as having said that Voldemort should have killed Hagrid, but she needed him alive to carry Harry out of the forest. There isn’t a right or a wrong approach to this, but bear in mind that readers will see and think about character actions. They won’t necessarily even notice your carefully crafted symmetry.