Most shadow techniques create a very ``hard'' shadow edge; surfaces in shadow, and surfaces being lit are separated by a sharp, distinct boundary, with a large change in surface brightness. This is an accurate representation for distant point light sources, but is unrealistic for many real-world lighting environments.
An accumulation buffer can let you render softer shadows, with a more gradual transition from lit to unlit areas. These soft shadows are a more realistic representation of area light sources, which create shadows consisting of an umbra (where none of the light is visible) and penumbra (where part of the light is visible).
Soft shadows are created by rendering the shadowed scene multiple times, and accumulating into the accumulation buffer. Each scene differs in that the position of the light source has been moved slightly. The light source is moved around within the volume where the physical light being modelled would be emitting energy. To reduce aliasing artifacts, it's best to move the light in an irregular pattern.
Shadows from multiple, separate light sources can also be accumulated. This allows the creation of scenes containing shadows with non-trivial patterns of light and dark, resulting from the light contributions of all the lights in the scene.