"We don't know." That sentence started off my formal education in North American archaeology. The statement is related to the who and how of the first Americans. There are many theories of who the first Americans were and how they got here and the more evidence we find the less certain we are.
Many of us have been taught the Bering Land Bridge theory. This theory states that Siberian nomads crossed the Bering Land Bridge around 13,000 years ago and spread like wildfire all throughout the continents of North and South America. The dogma of this faith was Clovis technology. This culture, united in the technology of hunting tools designed to bring down mammoths, was believed to have been the first Americans who arrived in America around 13,500 years ago.
Problems existed however. First Clovis tools were found only in certain areas. There were no Clovis tools found in the possible passage way which the Siberians could have taken. Neither could they be found in Alaska or Siberia. Secondly, a clear passage way would have been a nightmare for nomads. Sure, all game animals would be in that narrow passage way but so would anything else with a tasting of meat (There were bears which could stand 12 feet high back then!). And finally as a nail in a coffin, there have been sites in Pennsylvania, Minnesota, and even Chile which pre-date Clovis.
So other theories have arisen to explain the settlement of the New World. A rising prominent theory of how the Americas were colonized is the Pacific Coastal Model. This states that Asians hugged the coast (now submerged) and migrated south. This helps explain why sites like Monte Verde in Chile and elsewhere along the Pacific Coast tend to be older than interior sites. Moving along the coast line would allow groups to obtain a large food supply without risking glaciated lands.
Other theories exist however and they have some evidence to back them up. Pre-Siberian American Aborigines is another possibility. Those who back this do not deny the Asian origins of modern Indians but claim there were humans here before that migration. Some of the oldest bones in North America have traits of Australians, Melanesians, and even Africans. It is thought that these small groups were one way or another absorbed into the invading post-Siberian/paleo-Indian population. The Olmec Indians of Mexico created giant heads which look like black Africans but evidence of African heritage is flimsy at best. Elsewhere, Polynesians got as far as Easter Island. Going the rest of the way to Chile would not be out of the question (and it appears some did and brought chickens with them).
Another theory, which I do not subscribe to but brings up powerful points, is the Solutrean hypothesis. Two well respected archaeologist proposed that Clovis technology is actually based on older European technology. Solutrean "culture" was in Europe from about 21,000 years ago until it completely vanished around 15,000 years ago. The theory says Europeans followed the edge of the sea ice to North America and settled there. Problems with this hypothesis are the impracticality of ancient men travelling along sea ice in the middle of the ocean going as far as the Americas (sea ice does not make a good camping site). Also, why the gap of several thousand years between European Solutrean and American Clovis? There should be some overlap.
The Solutrean hypthoesis has one trump card: Haplogroup X. Primarily a European genetic trait, 25% of all Algonquian people have the genetic marker. The sheer number and background checks suggest this is way too high even with the mixing of American Indians and Americans of European descent. Add onto the geographical eastern bias of Algonquians and it seems there reasonably could have been a European migration which mixed with post-Siberian paleo-Indians.
The main history still stands. Genetic tests and physical traits like shoveled teeth still point to an Siberian Asian origin for American Indians. This population was the first one to successfully settle and dominate the Americas. More would come though. The Eskimos came latter and began a bloody campaign to dominate the Arctic. Groups came and went. Some lived in cities which rivaled those of the classical Old World, some lived in cities which were on par with the ancient ones of Mesopotamia, some became farmers in villages, some reached the point of semi-settled but remained partly nomadic, and others like the Sioux were the equivalent of the Barbarian Hordes of Europe. They lived and left there mark on the New World which in part last today. Many place names still carry their memory.
Even with pre-Columbian contacts, things evolved on an independent path for the most part. Then a group of men in 1942 led by an Italian under a Spanish flag would change the whole world.