DRI is Fred Miranda's approach to (and software for) HDR.

There was a time when high dynamic range images were spoken of in polite company. Essentially, it just means any method of combining different exposures to capture more detail in the shadows and highlights of a finished image than you could have gotten with a single exposure, and it was probably most commonly used in astrophotography.

Over the past few years, the term "HDR" has become associated with heavily tone-mapped images displaying flat, often desaturated colors, a lot of textural detail emphasis, and "halos" around areas of great tonal difference. HDR images don't have to look like that, no matter what software you use, but the overprocessed look has become something of a fad. The name "DRI" is there to differentiate what Miranda's software does from the fad look.

You can achieve high dynamic range images that don't look overprocessed in a number of ways. This article over at Luminous Landscape describes a few of them; I generally use multiple, separately-processed layers and layer masking (using Topaz ReMask, though I have used OnOne Perfect Mask and Corel Knockout in the past).