Anywhere the Density Shader is >0, the density of the cloud at that point will be >0, proportional the value coming from the Density Shader. The actual density is further affected by "edge sharpness" and "cloud density" in the following way: it is multiplied by "edge sharpness", then clamped so it never exceeds 1, and then multiplied by "cloud density".
Both "edge sharpness" and "cloud density" have the ability to change the density at the edges of clouds. However, because of the clamping, only "cloud density" can change the density everywhere.
Regarding so-called god rays, the reason these two settings affect them is because they affect the density of the medium, therefore affect the apparent visibility of the medium, which affects the visibility of the god rays.
Matt