How does UV affect skin colors in dark-skinned people? Skin color is one of the things one would rather not ask anything about! Only in humans, it can vary from very dark brown to pale pink. In darker-skinned people, the color is mainly due to melanin, which is produced by melanocytes. 
There are three types of melanin: eumelanin, pheomelanin, and neuromelanin. Eumelanin is the most common; so it must be the reason of the dark skin of the dark-skinned. And that's true, since the two major groups of eumelanin are black and brown species.
The first ray produced by sun that comes to mind when you think about affecting the attributes of skin is UVA, since it penetrates most into the skin (Even though its energy is lesser than UVB). So, when I attach these two together I reach the fact that if UV rays do not cause free radicals they will eventually change the skin's color. (This is what is mostly believed and its effect on natural selection is impossible to hide)
Now, I give up. How does UV do to the skin that makes it darker? This must be very easy, huh? What am I missing?
There is a journal article almost exactly about the title question, Mechanisms of Skin Tanning in Different Racial/Ethnic Groups in Response to Ultraviolet Radiation Journal of Investigative Dermatology (2005) 124, 1326–1332.  

However, the body of the question assumes that eumelanin changes color.  Instead, the article finds that melanin gets closer to the surface of the skin, and the degree to which this effect occurs varies with race/ethnicity.