The Pixel Shader allows for things like Lighting effects, Shadows, Bump Mapping, Texture Morphing, and Alpha Blending between multiple textures. Basically you are able to do any texture blending operation you can imagine. Another advantage to using ATi's Pixel Shader is that all these operations are done with very few rendering passes. Other hardware will redundantly use textures and go through multiple passes to create the safe effects. In other words the ATi implementation can give you all the visual benefits with very little performance loss. The one big thing to note here though is that DX8 is needed to take full advantage of this feature.
Above is an example of a shading operation done in DX7. This method has its limitations, as it cannot offer more efficient use of the same textures in one pass. Multiple passes must be made to create given effects.
Here we have an example of what the Radeon256 can do when coupled with DX8. This method is far more complex and allows not only efficient use of a single pass but gives developers greater control of the effects they may create.