Growing up in the late 70s and the 80s, I was fed a diet of media in which the native peoples of the United States were referred to as "Indians", while simultaneously being told in school that the proper, respectful term was "Native American." The term "Indian" of course comes from mistakes that early European settlers had made regarding where they thought they had landed. They believed that they had found the islands then known as the Indies, hence the people living on them were Indians. Even after they realized that this was incorrect, the term "Indian" stuck for quite a long time.
Still, as a kid, I was informed that the proper term was "Native American", and any use of the term "Indian" in school was corrected. My grandmother, the one Native American person with whom I had regular contact, never voiced an opinion one way or another on the matter, so I went with it and used "Native American", feeling a bit uncomfortable when someone, usually someone older than I, used the term "Indian." Throughout college this continued to be the primary term used to describe the peoples of the Americas, and even into graduate school.
Then I began working with Native Americans on a regular basis. You can imagine my shock when they referred to themselves, consistently and with few, if any, exceptions, as "Indians." How they reacted to me calling them "Native Americans" varied greatly - most don't react one way or another, a few react negatively (I have been told on a few occasions "no, we are INDIANS!"), and a few react positively (feeling that the use of the term "Indian" is fine for themselves and their fellows, but those of us who they do not recognize as being part of their community should use the term "Native American" or, where I work, "Native Californian"). It's probably little surprise that I took to using more specific terms (referring to people as Chumash, Yokut, Miwok, Ohlone, and so on, terms which never seem to elicit a negative reaction).
What accounts for the continued use of the word within Native American circles? Well, ask ten different members of the community,a nd you'll likely get ten different answers as to why they still use the term, but there are a few common reasons that I have come across. One is that the term was long something of a term of abuse (hence the reason why Native American became the more polite term), but through using it and claiming it, many people feel that they have taken the sting out of it. Another related reason that I have encountered is that many people within the native communities feel that they have earned the term through centuries of being saddled with it and having to deal with European views of all native peoples being pretty much the same, and always treated as inferiors. Another reason that I have heard is that each group had their own term for themselves, and that they consider the term "Indian" and the term "Native American" to be equally arbitrary, and are simply a bit annoyed at the desire of non-natives to switch terms.
Regardless, watching this play out has been rather interesting, and has provided a valuable lesson in how different individuals and groups place often wildly different values on the ethics and politics of how ethnic terminology develops.
I have also learned that few people, if anyone, minds when you refer to their particular group by its proper name. So, that's probably the biggest practical lesson here.