The word Indian is for people from the country of India. The word indigenous is used to describe native peoples of the land who lived there before the arrival of colonizers.
However, some people use "Indian" for indigenous people. It is not correct, but people do it anyway - this is because the earliest European colonizers of the Americas mistakenly called the native people "Indians."
Members of the indigenous peoples of the United States can be called Native Americans.
You can read more about the history of names for native peoples here: http://en.wikipedia.org/wiki/Native American name controversy