What is "the shortest half of the data"? Here is a histogram (realized with JMP) displaying two types of box plot called outlier box plot and quantile box plot. 

Right below, there are a bunch of explanations of the meaning of the different features of the outlier box plot.

The red bracket is "the shortest half of the data (the densest region)". What does this expression exactly mean? How different is it from the quartiles?
Source
 A: Think of all the ways that you could split the data into 2 "halves", You could start with the minimum to the median, then go from the 2nd lowest point to the point just above the median, then 3rd lowest point to ... then from the median to the highest point.  Measure the length of all the "halves" that you computed and the red bracket shows the one that has the smallest range.
A: The red bracket represents the smallest interval that encompass 50% of the data.
The two quartiles represents the first 25% of the data from each side of the mean encompassing for 50% of the data but this is not necessarily the densest region.
Imagine a skewed distribution that has a very long tail on the right. The median might be much higher than the mode and be fairly high. In consequence the 25% quartile will eventually include the mode but the right quartile will be long in a very sparse space. The densest region necessarily include the median (exception in @whuber's comment below) and one of the two quartile.
A: The JMP documentation is a little unclear--it just cites a book: Robust Regression and Outlier Detection.
However, it looks like the bracket contains the "highest density interval", or "highest probability density" interval. This is the smallest (i.e., shortest) interval that contains 50% of the data points.
Suppose your data looks like this

-30, -20, -10, 1, 2, 3, 4, 5, 6, 10, 20, 30.

The 50% densest region runs from 1 to 6 because it contains half the data (six of the twelve points) and is the shortest interval that does so: it's five units long, while the next closest interval (2 to 10) is eight units long instead.
The highest probability interval often shows up in Bayesian settings, sometimes as an alternative to confidence intervals. Here, they probably mean for you to use it a heuristic for detecting outliers.
