There has been a significant amount of debate on the pre-contact population of the Americas. The best pop-introduction that provides a surprisingly good summary of the debate is Mann's 1491. Denevan's The Native Population of the Americas is an older (1992) more scholarly approach, but still readable.

The main place of conflict is on the island of Hispaniola, what is today the Dominican Republic and Haiti. David Henige was the major instigator of a debate over the population on this island in 1978 with his essay "On the Contact Population of Hispaniola: History as Higher Mathematics," (Hispanic American Historical Review, pp. 217-237). In his essay, Henige claims that most any estimate of population on the island is impossible, as historians have taken to estimating the population based on next to nothing. Zambardino wrote a reply a few months later (HAHR, 1978 pp. 700-708) where he argues that estimating the population of Native America both has value, and that the estimate of 1,000,000 is a reasonable estimate for Hispaniola. Most historians following the debate have tended to agree with aspects of both of their arguments, but the estimate of 1,000,000 for Hispaniola in 1491 is reasonable. The remarkable element once you consider this as the pre-contact population, is that the more acceptable numbers in the early sixteenth century are that much more striking. The generally accepted count of the number of Native Americans on Hispaniola in 1514 is 30,000 (Henige, p. 218).

Estimates for the rest of the Americas are even more speculative than that for Hispaniola because they do not have the level of source material necessary to make accurate estimates. For Hispaniola, we at least have the Spanish documents of the first two decades that give some clue as to what was happening on the island. For North America, the diseases and death traveled across the landscape ahead of Europeans, depriving both the Native Americans of their lives, and historians of fist hand historical accounts of the population.

Deneven in 1992 estimated that there were approximately 50,000,000 Native people in the Americas in 1492 (Native Population, xxix). This estimate is likely too high for most historians, as his estimates for (among others) Mexico and the Andes are considered to be too high at 13.8 and 11.7 million respectively. More modest estimates point towards a total population of 25 million for all of the Americas (Mann, 2nd ed. p. 113-114). Either way, the estimate for the number of Native Americans living just 60 years after first contact dwindles to approximately 50,000-100,000. That is between a 96-98% population reduction over the course of 60 years, across two continents. So whether the population of North America was 10,000,000 or 50,000,000 (Dobyns " An Appraisal of Techniques with a New Hemispheric Estimate" *Current Anthropology, 1966, pp. 395-416), scholars accept that Native Americans experienced a demographic collapse of 90% or more of their population.

Edit: Also see this essay by Russell Thornton (published in American Studies, 2005) for a good introduction that's free online.