What you're talking about is called Motion Blur.  An Easy way to produce this is to use a directional blur, found in most paint programs.  Simply duplicate your layer, and apply a dirrectional blur filter to the lower layer.  Viola!  Motion Blur.

If you are talking about doing this to an animation, then you will need to use a program such as Aobe After Effects or Vegas Video, which have motion blurr capabilities.