Direct Rendering Infrastructure
Basically, DRI is yet another hack to support some hardware. Before, the Linux people had SVGALIB to get graphics, which did the work but was not good; now we have similar problem with GLX. DRI is not really nice, but one of the things that almost work.
DRI allows, to some extent, X programs to do OpenGL rendering directly on hardware. The actual drivers are built as kernel modules. DRI is one of the ways in XFree86 4.0 and later to get the GLX support. In addition to the correct GLX driver module for XFree86, you also need the appropriate kernel module.
It should be noted that DRI is not the only way to get the support. The Utah-GLX project is working on a less-hacky-sounding version of the interface. Also, NVIDIA's own binary-only drivers do the GLX support their own way. (This in case you're confused why typing "glxinfo" says "direct rendering: no" if you're using NVIDIA card, even when Quake 3 is working really fast and nicely.)
Home page, as of writing: http://dri.sourceforge.net/
Thanks to fuzzie for more explanations and clearing of confusion...