Diffuse Inter-Reflection is the process that allows light to be diffusely reflected off non-specular surfaces.
In real life terms what this means is that light is reflected off non-shiny surfaces like walls etc. to reach areas not directly in view of a light source. If the diffuse surface is colored, the effect is that of color-bleed, meaning that colored diffuse light will alter the color of surfaces nearby.
In computer graphics terms we call this global illumination.
There are a number of ways of achieving diffuse inter-reflection. Radiosity and distributed ray-tracing are the most common.