Hi, this question came up among friends and in my university class this semester and I’ve heard a few different takes.
To preface, I'm familiar with the fact that the term “Indian” derives from early reports sent back by European settlers of the inhabitants of North America when the settlers believed incorrectly that they had arrived in the West Indies after sailing from Europe. With that in mind, I understand that Indian is the “incorrect” name insofar as the inhabitants of North America were actually members of various tribes, not inhabitants of India.
I have heard arguments that the term “Native Americans” is more accurate and corrects the mistake that early European explorers made. I’ve also heard that ultimately, the best approach is to refer to these individuals most accurately by calling them by their tribal names; whether Comanche, Cherokee, or Powhatan, etc, and that a broader term is not appropriate. I’ve also heard arguments made by some that it is the right of the descendants of these cultures to decide which name best describes their history; whether “Indian”, “Native American”, or otherwise.
My university professor (an individual with amazing credentials, numerous well-regarded publications, and a high degree of respect in the historical community) suggested that the term “Indian” is a misnomer but has unfortunately been adopted by history as the correct name for this particular group of people - that is, the inhabitants of North America prior to settlement by Europeans. In her view, the Cherokee or Powhatans of the 17th century would be no better served by a term such as “Native American” seeing as “America” itself is a name given by some Italian merchant who happened to first draw the continent on a map. She argues essentially that history doesn’t care if we may prefer some different term than “Indians” today. The “right name” in this case just happens to be the product of a mistake.
Is she right? Would love any feedback, context, or clarification. My professor’s take struck me as somewhat insensitive but I’m not certain that such an opinion isn’t warranted. The fact may be that “Indians are Indians”, plain and simple, regardless of the mistake early European explorers may have made.
There have been a number of answers to this question in the past. Take for instance /u/Snapshot52's excellent answer on this.
The general guidance as it stands today is that "indigenous" is the term to use when referring broadly to the aboriginal inhabitants of the Americas. The phrasing "Indian" on its own is considered archaic, but the term "American Indian" is still widely accepted in the US, as demonstrated by the Smithsonian's Museum of the American Indian. "Native American" is likewise often acceptable, but generally doesn't encompass [northern] Alaskan and Candian indigenous groups who have other terms.
However, the most specific term available should be used whenever possible and if the subject prefers a particular label, use it. In my experience it's fairly rare to encounter situations where "indigenous" or "American indian" is the appropriate descriptor to use.
To add to what /u/AlotOfReading says, there's another answer from /u/snapshot52 that explains why Indigenous people are still often referred to as "Indians." You may also be interested in this portion of the FAQ from /r/IndianCountry, which goes into some detail on naming and terminology of Native people. In terms of what you or I should call someone who's Indigenous, the answer is that we should call them what they want to be called, but if you don't know, American Indian or Native American are terms that work in the U.S.
The context in which the word is being used can also make one term more appropriate than another. For instance, if a tribe self-identifies as “American Indian,” it would be appropriate to use that term to refer to them. Some tribes prefer the term “Native American” instead. And of course individual Native people all have their preferences. There really is not a “one-size-fits-all” term to use for all contexts. It should be noted, however, that “American Indian” and “Native American” do have political associations in the United States, whereas the term “Indigenous” does not. Being American Indian or Native American often is associated with tribal citizenship status. Those terms are not limited to political connotations, but it is good to be aware of them. The term “Indigenous” is more general in a sense. It can be used to refer to original inhabitants internationally as well.
In historical indigenous scholarship, the term the historian chooses to use is usually accompanied by a footnote that explains the choice of using that particular label. So, don’t feel like there is one right word and the rest are insensitive. The best way to navigate the terminology is to listen to the people in which they refer to. In conversation with a Native person? Ask them which term they prefer. Writing a paper about a particular tribe, Native person, or etcetera? Research how the tribe/person refers to themself or find out how relevant scholarship has referred to them.
It’s complicated and each term is certainly nuanced. So, use the term that best fits the context in which you are using it.