I could give a really long answer, but I think there are other treatments of the subject which are easily accessible. (Just do a web search with "chiasmus" and "Mormon" and tons of links will pop up, I'm sure.)

Basically, chiasmus is a type of Hebrew poetry which is found in both the Bible and the Book of Mormon, in which the poetical lines or topics are treated first in one order, then in the reverse order. This poetical form is normally used to highlight the point at the middle, so the central point of the discourse is also the central point physically. Some LDS members point to the presence of chiasmus (which was an unknown literary form in Joseph Smith's day) as a "proof" of the authenticity of the Book of Mormon. I hesitate to claim that, but the presence is at least "consistent" with the Book of Mormon's claim to be an authentic Hebrew document.

As an example, (this example is from Jeff Lindsey's page on the subject, http://www.jefflindsay.com/chiasmus.shtml) in the Book of Mormon, in Mosiah 3:18-19 it says: "...but men drink damnation to their own souls except they humble themselves and become as little children, and believe that salvation was, and is, and is to come, in and through the atoning blood of Christ, the Lord Omnipotent. For the natural man is an enemy to God, and has been from the fall of Adam, and will be, forever and ever, unless he yields to the enticings of the Holy Spirit, and putteth off the natural man and becometh a saint through the atonement of Christ the Lord, and becometh as a child, submissive, meek, humble, patient, full of love, willing to submit to all things which the Lord seeth fit to inflict upon him, even as a child doth submit to his father."

Jeff wrote it out like this, so the chiastic structure is evident:

(a) They HUMBLE themselves

(b) and become as little CHILDREN

(c) believing that salvation is in the ATONING BLOOD OF CHRIST;

(d) for the NATURAL MAN

(e) is an enemy of GOD

(f) and HAS BEEN from the fall of Adam

(f') and WILL BE forever and ever

(e') unless he yieldeth to the HOLY SPIRIT

(d') and putteth off the NATURAL MAN

(c') and becometh a saint through the ATONEMENT OF CHRIST

(b') and becometh as a CHILD

(a') submissive, meek and HUMBLE.

That's a very good chiasmus, even though the central part is not particularly being highlighted.

However, sometimes I think LDS members get a little too excited about claiming chiastic forms. John Welch, an expert on the subject, gave a list of criteria to be used in determining the

"degree of chiasticity" for a given work. (FARMS, Journal of Book of Mormon Studies Vol.4,

No.2, p.12.) Here's my summary:

1. Objectivity - it the structure obvious to an objective observer?

2. Purpose - it there an identifyable reason why the author would have

chosen to use chiasmus?

3. Boundaries - does it start and stop at natural barriers as opposed to arbitrary positions?

4. Competition with Other Forms - Is there a different literary device whichh would also explain the chaistic pattern (example here is "Hickory Dickory Dock" = limerick)

5. Length - the larger the number of chiastic elements, each of which bears its own weight, the "stronger" the chaismus.

6. Density - Tightness in the text is indicative of greater craftsmanship. I.e., proposed chaisms covering the entire book of Mosiah are les dense than Helaman 6:10.

7. Dominance - Are all the dominant nouns/verbs contained in the proposed construction?

8. Mavericks - are key words excluded from the proposed construction because they don't "fit"?

9. Reduplication - If the same word/element appears many times, the likelihood is that some non-chaistic force (including random repetition) is responsible. E.g., Alma 36 contains 201 words that apear once or twice, 58 words that appear 3x, 4x, or 5x, and only 42 words that appear more than 5x, so reduplication is not a problem.

10. Centrality- does the chiasm have a well-defined center, or turning point?

11. Balance - is the number of elements on both sides of the proposed center nearly the same?

12. Climax - is the center of the chiasm something that the author would want to emphasize?

13. Return - does the end of the chiasm return to the subject of the beginning?

14. Compatibility - does the author have a history of using chiasm and/or related forms of parallelism, or is it an isolated case? (If not isolated, it's less likely to be coincidence.)

15. Aesthetics - Subjectively, did the author succeed artistically?

The more these items are present in a work, the more certain we can be an author used the chiastic form intentionally.

And that's actually where I think identifying a chiasmus is the most help to us-- so that we can better understand the ideas the author intended to transmit.