How far back do you want to wind the clock? Before about 100,000 years ago, everyone's ancestors were living in Africa. The ancestors of the Native Americans were on the doorstep of the Americas no later than 20,000 years ago (if not significantly earlier), having come up through western Asia and across Siberia. From here, some may have walked into Alaska if the sea levels were low enough at the time from the accumulation of glacial ice during the Ice Age. Others almost certainly took boats down the Pacific coast, penetrating into the interior of the continent at various points. Certainly by at least 16,000 years ago they had reached the east coast of North American and by at least 11,000 years ago they had reached the southern tip of South America. But remember this migration is constantly branching and intertwining as it goes, jostling against itself, as peoples split and merge again, continuously building new social identities as they go.

Sometime later (the exact time frame is uncertain) another major wave of peoples came across the Bering Strait from Siberia. These were the ancestors of the Na-Dene people, who mostly inhabit Alaska and western Canada, with some relatively recent groups migrating into the American Southwest within the last 1000 years (these are the Apache and the Navajo).

A third group, the ancestors of the Inuit, Yupik, and Aleuts, began coming across the Strait around 5000 years ago. They didn't come over all at once and they didn't forget about Siberia after they left, which likely serves as an example of how previous migrations went.The last big phase of their migration, the migration of the Thule (ancestral Inuit) began less than a thousand years ago, though the Thule's ancestors had come over from Siberia a bit before that (their big push was out of cultural centers in northern Alaska).

The Inuit-Yupik-Aleut languages form a large linguistic family, like the Indo-European languages in western and southern Eurasia. The Na-Dene languages do as well, though there are a few languages on the periphery whose inclusion is debatable. The Haida are a good example here. Is it a divergent Na-Dene language; if so, did it diverge after the Na-Dene people came to the Americas or did it diverge before and come over in a separate migration of its own (smaller migrations like this could easily go undetected in the archaeological record)? Is it an unrelated language that happened to borrow elements and vocabulary from the neighboring Na-Dene languages?

When you get to the first big migration, it happened so long ago that any continent-spanning linguistic connections have been buried by the millennia. The languages are broken up into several dozen major language families (with varying levels of proposed and vigorously disputed higher-order connections, such a linking Iroquoian and Siouan languages into Macro-Siouan or Algic and Salish languages into Almosan). Not only do are the hypothetical roots linking the big language families together obscured, there are many language isolates that don't seem to fit in with any of the major language families; their speakers branched off from the main linguistic trunk so long ago that their language is no unlike any of spoken by their neighbors.

Now if you dig into this topic enough, you'll inevitably come across some theories outside the anthropological consensus. We can set aside some outlandish ones, such as Native Americans being descended from Israelites or the Welsh. I'll mention two though that at least attempt to be scientific about it. The first is known as the Solutrean Hypothesis, and it suggests that, in addition to the peoples already discussed, another wave of came across the Atlantic during the Ice Age from Western Europe, and this explains the 16,000 (or earlier) settlements on the east coast of North America. I want say more about that here, because I've already discussed the problems with it elsewhere.

Another idea proposes a fourth -or rather a new first - migration into the Americas. This hypothetical migration is part of the same migration of people that populated Australia by at least 50,000 years. After reaching southeast Asia, some continued on into Australia while others headed north, eventually crossing into the Americas before the previously mentioned "first" migration. Like the Solutrean Hypothesis, this hypothesis has very few supporters. It's main proponent is Walter Neves, who's work has focused on Paleoindian remains, mainly from South America, which he claims has morphological features more similar to Australians than most South Americans. That recent indigenous populations in South America had similar features is seen as evidence that the two founding populations mingled (see " Recovering mitochondrial DNA lineages of extinct Amerindian nations [...]" by Gonçalves, et al). Neves morphological studies (and similar studies by a few others, like the ones that assigned Kennewick to European or Polynesian peoples) have been criticized on various grounds: namely that 1) Native American peoples are physically more diverse these studies presume and 2) so-called "racial" morphological markers are not static in time, and the picture is complicated by as lineages diverge and merge again. Still, new genetic data is coming out all the time, as it did in the previously mentioned Gonçalves' study, and it's occasionally anomalous. I just noticed that Gonçalves and friends published a further analysis of that unknown mtDNA last year, in which they identify it as a Polynesian lineage. I'll have to read that later. If it is, in fact, a Polynesian mtDNA lineage, that actually hurts Neves' hypothesis far more than it helps it, as the proposed Polynesian contact event occurred much, much later and involved a different population than Neves' hypothetical migration.

It is unlikely that anyone crossed Beringia during the last Ice Age. Current theories support that there were likely tundra plants and some animals on Beringia during its ice-free periods, but Canada is a completely other story. There would need to be some miraculous corridor or corridors in Canada (for which there is no evidence) where people could have passed southward to the continent. The northern third or so of North America was massive glaciers, and this was true up until the reopening of the Strait.

What is likely, however, is coastal migration, not overland. Heading along the edge of Beringia would have allowed ample fishing opportunities. Of course, the vast majority of islands they ran into along the way are underwater now, and what they might have left by way of stone tools and waste would be quite difficult to find. But it's a hell of a lot more likely then trudging across the glacier to America. This paper is a good resource.

There, as well, another and rather divisive theory. This page has a cute map and references some of the supposed evidence. The issue isn't whether or not it's technically possible - it certainly is - the issue is more about whether anyone did. It would have been very possible for settlers from Polynesia to reach Chile and vice versa. The Polynesians are very accomplished sailors. But that's quite a bit of open ocean and would have required quite a few people to sail across it.

The DNA evidence is pretty unsure, too, more and more papers are coming out about the supposed DNA similarities and the evidence doesn't strongly support a Polynesian origin. Sweet potatoes, however, might be a bit more compelling. And, in the interest of elegance, the Polynesian origin explains the pre-Clovis Monte Verde site better than the Bering coastal one.

EDIT: A Polynesian origin would certainly not exclude a coastal Beringian origin, which I'm not sure if I made clear.