The name by which it is polite to call a group of people is sometimes in flux and changes over time as political and social perceptions change. What was once normal can become offensive over time, usually because it was always offensive to begin with.
The name of peoples whose ancestors were originally in the Americas before colonization is a tricky subject, mainly because they were never lumped together in one group before. Each tribe or family group called themselves something different. As such, today each group still prefers different names. We discuss to commonly accepted terms below, but if this group is in your audience, it is best simply to ask what they prefer. Many will say they prefer the specific tribal name they subscribe to.
Indian was the common term for native peoples for a long time. It stems back to Christopher Columbus. He called the people gente en Dios or people of God, which morphed over time to Indios and then Indian. Some people still prefer to self-identify with this name. However, as more mixture happened, especially with people coming from India, the name became confusing. In fact, in the dictionary its primary definition now is a person from India. Whereas the original definition has become secondary.
American Indian is the preferred politically correct term currently. It refers to the same group of people, but again mostly from the continental United States. Alaska has Alaskan Natives and Eskimos (two different groups). There are Native Hawaiians and more. In Canada the term is Canadian Indian or First Nations.
Native American is a noun referring to persons that are part of an indigenous group. It is mostly associated with the United States, even though the name should include all of the American continents (more about this difference here). This term was revised mainly because it too was unclear. Anyone born in America is native to America. In some areas close to American Indian reservations it is common to hear American Indians referred to as natives. This is derogatory and should be avoided. As should the terms ingin and redskin, though that should be common sense.