"Native American" is the term currently used by the government of the United States of America for the indigenous peoples under its dominion. The government of Canada calls its own indigenous subjects "First Nations." Ordinary people, unburdened by the fetters of political correctness, will often use the older term American Indians for the indigenous peoples of North America collectively.
But this usage, conflating "native" with "indigenous" or "aboriginal," contradicts both common usage in the rest of the world, and the history of the term "Native American" itself.
In the nineteenth century, a Native American was understood to be a person of Western European stock whose ancestors had been in the New World for several generations. The meaning was similar to criollo as used in the Spanish-speaking regions of the Americas. It was a broader term than WASP - a Native American might trace his ancestry to Scandinavia or the Low Countries - but it did exclude the swarthy descendants of newer immigrants from sunny places. A 19th-century American blue blood, raising a ruckus about the boatloads of shiftless wops and kikes despoiling the shores of the land his forefathers conquered, would see a grotesque irony in his own term of ethnic pride given to a bunch of redskins.
In general, if X is a politically unified territory, the native people of X are those who originally conquered and unified it, and not any subjugated nations or tribes that may have preceded them. The Basques are not "Native Spaniards," the Ainu are not "Native Japanese," the Lapps are not "Native Norwegians," and the Etruscans most certainly were not "Native Romans." When Englishmen first landed at the future Virginia and the future Massachusetts, they didn't meet any Native Americans. The people they met had never heard of anything called "America." But when other Englishmen landed at Hawai'i, they found a Kingdom of Hawai'i populated by Polynesian people who called themselves Hawaiians. Hence the descendants of those Polynesians rightly call themselves Native Hawaiians to the exclusion of the English descendants, who are merely haoles.
There probably isn't a truly accurate term for North American Indians in existence. But the Canadian political moniker "First Nations" is certainly fairer than the USA's alternative. It recognizes that those peoples, regardless of whether they were ever really "nations," are indeed distinct from the British and French Canadians who conquered and unified the country. And it encourages honorable treatment of them - it looks far worse when a government breaks a treaty with a conquered nation than when it merely displaces and abuses people who, having been officially defined as "natives" of the conquering empire, are left without either the security of their own heritage or the benefits of true assimilation.
Of course, there's nothing new about the USA's rulers manipulating language for purposes of political dissimulation. The conquest of seceding states whose rulers merely wished to go their own way was passive-aggresively redefined as a "civil war." The doctrines of the corporate-statist coup that was the New Deal became "liberalism." And now, the USA works to define its American Indian subjects out of existence by calling them "Native Americans."
In the coming decades, as the empires of the West implode, many Americans and Europeans may find that their homes now sit on land ruled from Beijing. I suspect I'll be able to tolerate life as a Chinese subject - Beijing can't possibly be much worse than Washington at this point - but I might just blow my top if some officious bureaucrat insists that I'm a "Native Chinese."