The introduction to the Book of Mormon tells of 3 groups of people (well, 4 actually because it makes reference to Columbus as well) who traveled from the Old World to the American Continents. The first group were the Jaredites who came from the Tower of Babel when God confounded the languages of the people, somewhere around 2000 BC. This civilization destroyed itself in civil war a few hundred years before Christ. The second group was Lehi and his family who left Jerusalem around 600 BC. They divided into two groups once they got here, one being the Nephites and the other the Lamanites. There was also the family and friends of Mulek who fled Jerusalem after it was destroyed and made their way here. The ended up meeting with the Nephites and joining with them. (They had also met briefly the last living Jaredite before he died.) The Nephites and Lamanites were often at war. About 400 years after Christ, the Lamanites finally destroyed the Nephites. It is understood that these Lamanites are the principle ancestors of the Native American Indians. 
Whether there were other migrations of people to the Americas who might also provide ancestry to the Native Americans, the Book of Mormon does not state.

The LDS Church believes that the descendants of Lehi the Lamanites are among the popularly of the American Indians. At one time it was strongly believed that the Lamanites were the only ancestors of the American Indians but scientific and scriptural study has shown this not to be the case and most membership now agrees with the Church that the descendants of Lehi are one of many people who inhabited this continent from 600 B.C. forward.  
 
Lehi and his family was not the first to settle the America's, the Book of Mormon also tells us of another people called the Jaradites that settles this land around 2,000 BC, and later died off due to internal conflict. Many LDS scholars also point to the scriptures to show that the Nephites and Lamanites had other inhabitants that they neighbored with most likely members of the Aztec and Mayan tribes of Central America. While Israel has a connection to the Americas through the Nephites and Lamanites, they were not the sole inhabitants of the American Continents.

Because many scientists claim that DNA tests prove that Native Americans are in no way related to people from Jerusalem or Israel, but then I was thinking that if there were other tribes during the time when the Nephites and the Lamanites were in America then maybe they could have had children with members from other tribes thus mixing their DNA with other tribes and then through a course of years of interracial marriage would have made it much harder for scientists to trace the Laminates DNA.