This is always a contentious subject, because of what it implies for just how advanced or sophisticated Native American cultures were and also for what it implies for just how devastating the effects of European and African* immigration were.
There are two most widely cited works on the subject:
Atlas of World Population History by Colin McEvedy and Richard Jones (Penguin, 1978)
A Concise History of World Population by Massimo Livi-Bacci (4th ed., 2006)
From the dust jacket:
This book describes and explains the history of human population. It examines the changing patterns of its growth, and the effects upon it of migrations, wars, disease, technology, and culture. Since its publication in 1992 it has become the most widely read account of the subject...
It's an outstanding book, however, it does not cover the same territory as the earlier volume. As the little blurb tells us, it is largely a compendium of various theories of population growth and decline with special reference to Malthus, but also including many other points of view.
This work is so cautious in interpretating the data that Livi-Bacci can only bring himself to quote the work of others when giving them. For example, the most widely used table for population figures turns out not to be the author's, but rather, is lifted from the work of J.N. Biraben's Essai sur l'evolution du nombre des hommes in the journal, Population, 34 (1979).
There is very little detail, so if you're looking for population figures for a particular country at a particular time, this work will not be very helpful.
McEvedy and Jones' work, though written some years ago, is still the most widely cited work on the subject. As far as that goes, no other authors that I'm aware of have even attempted to duplicate their work. Other scholars have studied the problem in specific areas or time periods and those few who have attempted something more general don't actually disagree with the McEvedy and Jones book that much anyway. But the purpose of this thread is to share information. If someone has come across other sources that need to be examined, by all means, list them.
* Effects of immigration are usually thought of exclusively in terms of European colonization, but many of the diseases brought to the New World were from Africa, and as any cursory examination of the history of the Caribbean reveals, these were often as devastating to Europeans as to the indigenous population, and go a long way towards explaining the population make-up of the region today, largely descended from Africans, who had more natural immunity to these tropical diseases.