Image Based Lighting is a rendering technique where complex lighting is stored in a environment map which is projected onto the scene. In simple words, a light probe or environment map is just an image on a sphere. The following figures show an unwrapped environment map in the longitude/latitude format but other unwrapping methods are popular as well:
If the range of the image colors is within some small defined range (e.g. 0..255 for monitor display), the content is only LDR (low dynamic range). With HDR (high dynamic range) content some rendering effects will become more apparent and correct (e.g. DOF, motion blur, bloom, dark materials, global illumination). The preferred graphics card representation of such images is the cubemap. Depending on the content and compression requirements, various texture formats can be useful. The following image shows the 6 sides of a cubemap with it's mip-maps (useful for blurry reflections and to reduce aliasing):
Diffuse lighting can be approximated very well by convolving an environment map (diffuse convolution) which can be stored as a cube map again. Because of bilinear filtering, the texture can be quite low resolution. Mip maps are not required and the result with mip maps can actually look worse as ordinary mip mapping on the GPU is computed for each 2x2 pixel block and 2x2 block artifacts can become noticeable.
The following image shows how different materials react differently to the same lighting conditions.
The following images show IBL diffuse lighting only (top left), diffuse and IBL specular (top right) and diffuse and specular with added sun specular and diffuse (bottom).
Depending on the light probe, the lighting can be more ambient or harsh or even colored. The following images show some examples (including sun which casts shadow):