The brightness of a given water cloud varies a lot with which part of the cloud you look at, and whether you are looking at a side which is directly illuminated by the Sun or not. For instance, if you fly above the cloud layer, all clouds are bright white. Similarly, if you observe a large isolated cloud from a distance, its top is always bright while its bottom may be much darker.
So when you see a "black cloud", you probably mean that you are looking at the bottom of the cloud, from under it: in that case, sunlight would have had to penetrate the cloud from the top and find its way down until it is scattered in the direction of your eyes. There are many opportunities for scattering inside such a medium, and although absorption may be limited, scattering would take place in all directions. As a result, rather little light reaches you.
The deeper the cloud, the more water it may contain (although there are probably large variations, depending on cloud type, seasons, etc.) so it is not unreasonable to think that dark clouds (viewed from under it) might contain more droplets and possibly bring more rain. The correlation may still not be very strong, though, because it often happens that there are multiple cloud layers, so the observable darkness of the cloud may also be due to other factors.
Lastly, aerosols, and in particular smoke plumes or pollution, may exhibit different colors and brightnesses due to the chemical composition of their particles (soot, black carbon, etc.)
Thank you sir, Michel M. Verstraete for your explanation. As you say, when we observe clouds from top (from sun side) it is white and when we observe the same cloud from the bottom: earth's surface, it is black.As per an information, if thickness of cloud is more than 300 ft, it will appear black. Invariably, over 15-20 degree North latitude in India along west coast of Bay of Bengal or eastern coast of India, we observe most of the times, black clouds rain more. Also many times black clouds does not rain at all and disappear without any rain.
Yes, indeed, the formation of clouds is controlled by dynamic and thermodynamic processes (e.g., convection, condensation, latent heat release) and environmental variables (e.g., supersaturation, the presence of condensation nuclei, etc.), while rainfall depends on other mechanisms, in particular those that involve coalescing small droplets into larger drops, heavy enough to fall out, despite the inherent turbulence.
Clouds are highly dynamic: they can appear and disappear in a matter of minutes to hours, and of course not all clouds produce rain (or snow, or other forms of precipitation). Some of the rain that is generated can also re-evaporate in the atmosphere below the cloud, before it even has a chance to reach the land surface (this is called "virga": see https://en.wikipedia.org/wiki/Virga).