Although it is settled history that the American Indians are of Asian heritage, the time of their arrival and their arrival and departure points are matters of considerable debate. The safest conclusion seems to be that they may have arrived at different times and from different places.
The above answer notes the Clovis, New Mexico site which for many years was considered the oldest known settlement; however it has since been superseded by sites discovered at Meadowcroft, Pennsylvania and more recently in Monte Verde, Chile. The Monte Verde site, which has only recently been authenticated, has been radiocarbon dated to much older than the Clovis site. For this reason, there is some speculation that the Indians of Central and South America came in a separate migration from those of North America.
It is entirely likely that at least some paleo-Indians crossed the land bridge across the Bering Strait, commonly known as Beringia, although this is largely speculative. Since this land bridge only occurred during the last Ice Age, at least one scholar, C. Vance Haynes of the University of Arizona, has theorized that an ice free corridor developed in the Western United States which allowed free passage to the lower areas of the continent. There remains intense (and sometimes heated) scholarly debate on the earliest dates for arrival of peoples in the Americas.
An excellent discussion of this topic can be found in Charles C. Mann's 1491: America before Columbus. The link below also offers useful information.
Many of the details regarding this question are not known to us today. However, the basic outlines are.
According to scientists (many Native Americans have different views on this) American Indians originally came from Northeast Asia. They crossed over to the Americas in the area that is now the Bering Strait. It is not clear if they all came at times when there was a "land bridge" where the strait now is or if some of them came by sea. Some scholars believe that people could have come to the Americas by following the coast in their watercraft.
As for the date, this is the subject of great controversy. It is known that the "Clovis" people were in the Americas by 13,000 years ago. This used to be seen as the earliest settlement of most of the Americas. However, most scholars now believe that people were in the Americas before that. This may have been as much as 20,000 years ago.