What has the LDS Church (Mormons) historically taught about the Curse of Cain and dark skin?

I have heard that people with black skin had been denied certain things in the Mormon Church until the past 30 years or so and that this has something to do with the Curse of Cain. What has been the historical teaching of the LDS Church regarding the Curse of Cain and dark-skinned people? What are the references in the Book of Mormon and other Mormon Scriptures that address this issue?