Why are people called "Native Americans" or "American Indians"?
Names are labels for things, whether they are things, ideas, people or animals. We call things by the "name" the language we speak uses.
For English speakers, the word "cat" refers to the furry animal that goes "meow". In other languages, like Mohawk, the word is /tako:s/.
We are particularly interested here in names for different peoples. Some peoples are called different things in different languages. The Japanese call themselves Nipponjin, but Americans use a different term. The same problem exists for many of the native people of the Americas, and for the population as a whole.
American Indian versus Native American
The shift in names for peoples in different parts of the world is often a reflection of changes in political structure. The Belgian Congo is now Zaire. Since it is no longer under Belgian control, it has adopted this name. Many countries have changed name over time.
In a similar way, many ethnic groups have changed their names over time. The most common are the changes from "Colored" (still used in the National Association of Colored People) to "Negro" (as in the American Negro College Fund) to "Black" to "African American". Questions have been raised as to whether or not these terms mean the same thing or not. For example, African American might refer to people from Egypt who are now Americans and who are not "Blacks".
The term American Indian stems from Columbus' idea that he had found "India" rather than a "new" continent, and thought the people were Indians. Later, when it was realized there had been a mistake, "American" was added to the name in the U.S. and Canada, ("Red" in Britain) to distinguish the native populations here from the people who actually lived in India!
Some Native people have argued that the term actually comes from the phrase "in Dios" meaning "in God" since these were "natural" people. Indeed, the idea that many of the peoples encountered initially could be "naked and unashamed" implied that they were without original sin, since according to the Bible, Adam and Eve discovered they were "naked and ashamed".
There are some problems with this in that the word would more likely have become "Endian" since in Spanish one says "En dios" not "In Dios". Such problems can be circumvented in a number of ways.
A recent development has been the use of the term "Native American" (always spelled with a capital "N" to distinguish it from people who are native Americans [i.e. born in the Americas] as opposed to those whose ancestry is aboriginal). It has become the "proper" term according to some, and according to others it is simply "politically correct". Several prominent American Indians, like Russell Means and Wes Studi seem comfortable with "American Indian". Means has gone as far as to say it is a "government" term and not one normally used by Indians. Certainly when the term "Native American" first appeared that was clearly the case, although over time some politically active groups have argued for its use.
Since the various cultures in the Americas are all quite different, it would appear that no indigenous group ever thought of all the people as a single unit. This was something that developed as a result of contact and the Western established governments dealing with natives as though they were all alike. Governmental policy generally treats all natives the same, it would seem reasonable that the government would like a word to refer to the entire group of natives. Some Indians/Native Americans have said the whole thing is irrelevant since the term refers to a group of people considered the same by the Westerners, and not by any group themselves. This in effect asserts the negation of any term which merges the distinctive languages and cultural types together under one rubric.
It has also been thought that the term "Native American" is more inclusive than "American Indian". That Native American refers to any aboriginal population including the Samoans and Hawaiians for example, who as aboriginal populations in the United States fall under the category "Native American".
Names of Specific Peoples
The specific name of a given society may also be a point of contention. Some of the names are not the ones used by the people themselves, but are taken from the names other people called them. Apaches, for example seems to come from a word in one of the Pueblo languages that means "Enemies of the Cornfields". The Apache word is usually a variant of the Athabascan word for "people", "Dindeh".
In addition some variations in dialects have caused some difficulties in recognizing similar peoples. For example "Dakota" and "Lakota" are slight variations in the same word. "Chippewa" and "Ojibwa" are virtually the same word as well - the "ch" sound is related to the "j" and the "p" and "b" are also related.
Finally, early spellings were not consistent and as a result there may be several spellings of names that vary based on how well the person writing them down heard them, and chose to spell them. For example, the people known as "Cheyenne" had their name occasionally written as "Shian"!
While it isn't possible here to list all of the names of the different peoples and the names they are called, and call themselves along with translations, it can be said that many of the names translate as "People of the...", or "People Who...". Hence "ghat dindeh" is one of the words used for the people known as the "Kiowa-Apache" and translates roughly as "Cedar People". For the Iroquois, known variously among themselves as Rotinuhsesronu, or Hodeenasaunee the term has the meaning "People of the Long House".