How can some people say that dark skin is a curse from God? They point to the curse of Ham in Genesis 9.
The target of Noah's curse wasn't so much Ham as his future lineage, starting with his son Canaan. Canaan represents the inhabitants of the Promised Land (Canaan) up until the time of Joshua and Judges. There is, of course, no good reason to imagine that the Canaanites were more darkly complected than the Israelites.
Hatred and prejudice, in the eyes of the prejudiced, often look better when dressed up in religious garb. Now the Mormons are certainly not the only group who have justified racism on "scriptural" grounds (the South African Dutch Reformed Church comes to mind), but the Mormon scriptures do appear to reflect this thinking, especially common throughout the 19th century. (If you are skeptical, look them up here.)
In 2 Nephi 5, dark skin is a curse (though chapter 26 seems to allow that blacks are God’s children, too). Alma 3 has the curse of dark skin on Lamanites, although 3 Nephi 2 has them blessed, losing their dark skin and turning white. The book of Jacob too holds out hope that the dark-skinned will eventually be blessed.
Blacks are spirits who in their preexistence rebelled against God—their punishment was to inhabit black bodies. Some Mormon missionaries promised darker races they would be white in the next life if they convert. In 1976 a Mormon bishop in Vancouver was excommunicated for accepting a black into the Mormon priesthood. And IRS pressure seems to have triggered the 1978 “revelation” to President Spencer Kimball that it was time to allow the dark races into the ranks of the Mormon priesthood.
For more, see John Oakes and Douglas Jacoby, Mormonism: What Do the Evidence and Testimony Reveal? (Spring, TX: Illumination Publishers, 2012), and also the podcast on The Book of Mormon at this website.