The smallest region over which a density exceeds a threshold. More useful for summarizing multimodal distributions than other probability regions. Frequently used in Bayesian statistics ("Highest Posterior Density Regions"). The term in one dimension is "Highest Density Interval".
Here is the definition of the HDR, taken from Rob Hyndman's 1996 article "Computing and Graphing Highest Density Regions" in The American Statistician:
Let $f(x)$ be the density function of a random variable $X$. Then the $100(1-\alpha)\%$ HDR is the subset $R(f_\alpha)$ of the sample space of $X$ such that $$R(f_\alpha) = \{x\colon f(x)\geq f_\alpha\},$$ where $f_\alpha$ is the largest constant such that $$P\big(X\in R(f_\alpha)\big)\geq 1-\alpha.$$
Figure 1 from that article illustrates the difference between the 75% HDR (so $\alpha=0.25$) and various other 75% Probability Regions for a mixture of two normals ($c_q$ is the $q$-th quantile, $\mu$ the mean and $\sigma$ the standard deviation of the density):
The idea in one dimension is to take a horizontal line and shift it up (to $y=f_\alpha$) until the area above it and under the density is $1-\alpha$. Then the HDR $R_\alpha$ is the projection to the $x$ axis of this area.
Of course, all this works with any density, whether Bayesian posterior or other.
Here is a link to Rob Hyndman's page on HDRs. The article can be found on JSTOR Rob Hyndman's hdrcde package for R implements the algorithm from the papers and from subsequent work; Rob's page on the hdrcde package contains pointers to further literature.
