The Blur and Blur Offset controls determine how a 2D map is blurred, or how it is softened in the rendering. Their effect is visible only when you render final output.
Top: Blur = 1, Blur Offset = 0
Middle: Blur = 10, Blur Offset = 0
Bottom: Blur = 0.01, Blur Offset = 0.2
The Blur setting blurs the map based on its distance from the view. The farther away the map is, the greater the blurring. You should always use some blurring on your maps to avoid the type of scintillation, or aliasing that can occur when pixel details are reduced off in the distance. This effect typically occurs when you use detailed bitmaps viewed at a distance, and is particularly apparent during animations. The Blur default is 1.0, which is a good setting for most purposes.
Blur Offset blurs the map without regard to depth. That is, all the pixels in the map are blurred equally, regardless of how close or how far they are from the camera.
Whereas the Blur value is primarily used to avoid aliasing, Blur Offset is useful when you want to soften, or defocus the details in a map. It's the equivalent of blurring the bitmap in an image-processing program before applying it as a material map.