It seems that the term is almost exclusively used to describe pre-colonial Australians, but most definitions that I found don't specify any given ethnicity.
This is more a matter of custom, fashion and etiquette than meaning.
Native American means per M-W:
a member of any of the indigenous peoples of the western hemisphere; especially : a Native American of North America and especially the U.S. — compare American Indian
"Native American" while the preferred usage is potentially ambiguous particularly is varied for example to "Native Coloradan" which can refer without capitalization to someone born in Colorado, but with capitalization to someone who is a Native American who lives in Colorado.
Also, the term Native standing alone rather than qualified to refer to indigenous people is often considered mildly derogatory. Hence one would avoid saying: "The natives consider corn, beans and squash their traditional foods."
American Indian is a mildly outdated term that means more or less exactly what Native American does.
There can be dispute over whether the indigenous people of Hawaii or Alaska count as "Native American" (they do not for U.S. Census Bureau purpose), and it would actually be quite rare to refer to people who are indigenous to the Americas south of the United States as "Native Americans" unless they are part of a tribe or culture that existed both to the North and to the South of the current Mexican border.
According to the U.S. Census Bureau for the most recent 2010 Census (the definitions of racial and ethnic categories have shifted somewhat in almost every census):
The concept of race is separate from the concept of Hispanic origin. Percentages for the various race categories add to 100 percent, and should not be combined with the percent Hispanic.
White. A person having origins in any of the original peoples of Europe, the Middle East, or North Africa. It includes people who indicate their race as "White" or report entries such as Irish, German, Italian, Lebanese, Arab, Moroccan, or Caucasian.
Black or African American. A person having origins in any of the Black racial groups of Africa. It includes people who indicate their race as "Black, African Am., or Negro"; or report entries such as African American, Kenyan, Nigerian, or Haitian.
American Indian and Alaska Native. A person having origins in any of the original peoples of North and South America (including Central America) and who maintains tribal affiliation or community attachment. This category includes people who indicate their race as "American Indian or Alaska Native" or report entries such as Navajo, Blackfeet, Inupiat, Yup'ik, or Central American Indian groups or South American Indian groups.
Asian. A person having origins in any of the original peoples of the Far East, Southeast Asia, or the Indian subcontinent including, for example, Cambodia, China, India, Japan, Korea, Malaysia, Pakistan, the Philippine Islands, Thailand, and Vietnam. It includes people who indicate their race as "Asian Indian," "Chinese," "Filipino," "Korean," "Japanese," "Vietnamese," and "Other Asian" or provide other detailed Asian responses.
Native Hawaiian and Other Pacific Islander. A person having origins in any of the original peoples of Hawaii, Guam, Samoa, or other Pacific Islands. It includes people who indicate their race as "Native Hawaiian," "Guamanian or Chamorro," "Samoan," and "Other Pacific Islander" or provide other detailed Pacific Islander responses.
Two or more races. People may have chosen to provide two or more races either by checking two or more race response check boxes, by providing multiple responses, or by some combination of check boxes and other responses.
Despite this guidance, empirically, the vast majority of Hispanic people who are Mestizo (see below) report their race as "Other" rather than "Two or more races" listing both European and Native American as the Census Bureau which the Census Bureau would usually consider to be correct.
As noted in the comments, the term First Nations is the preferred term for the indigenous peoples of Canada who are not Inuits or other indigenous Arctic people.
Historically, Inuits were called Eskimos (although Inuit is a somewhat more exclusive term than Eskimo that excludes the indigenous people of the Aleutian Islands of Alaska also known as Native Alaskans. Eskimo is a term that has fallen out of favor for no obvious reason other than that this is not what members of that group of people call themselves, and not because Eskimo was particularly derogatory when it was in use. The historical archaeological culture that was ancestral to and is in continuity the modern Inuit is called the Thule (the Thule are not Paleo-Eskimos).
Arctic peoples of Canada and Alaska who have since died out and been replaced by Inuits are still called Paleo-Eskimos in the academic literature and sometimes even in the popular press without offense (in general, offending a group of people who have died out tends to be harder to do than offending a group of people who still have living representatives). Paleo-Eskimos would include the Dorset archaeological culture and the Saqqaq cultures of Arctic North America.
Latin America has its own set of terms to describe persons of indigenous or partially indigenous origins, the most important of which is Mestizo which means a person of combined European and indigenous American (also called Amerindian descent). Many people use the word Hispanic not in the sense defined by the U.S. Census Bureau but to refer to someone who appears to be Mestizo.
"Aboriginal", "indigenous" and "autochthonous" have very similar meanings.
indigenous is used in both technical and non-technical writing and has fewer connotations than most other terms of the type. It means per M-W in the relevant sense of the word:
1: produced, growing, living, or occurring naturally in a particular region or environment * indigenous plants * the indigenous culture
Aboriginal is usually used in common speech and capitalized to refer to indigenous Australians or by anthropologists in a mildly outdated choice of words to refer to indigenous people. It means per M-W:
1: being the first or earliest known of its kind present in a region * aboriginal forests * aboriginal rocks
2: of or relating to the people who have been in a region from the earliest time : of or relating to aborigines * aboriginal languages * aboriginal tribes/customs/art; specifically, often capitalized : of or relating to the indigenous peoples of Australia
autochthonous is almost always used as a technical term by anthropologists. It means when referring to people per M-W:
1: indigenous, native * an autochthonous people * autochthonous plants
In some cases, anthropologists will use the term autochthonous to distinguish.
For example, between indigenous people of the Americas who are descended from the founding population of the Americas, and indigenous people of the Americas who arrived many thousands of years later such as the Thule and Paleo-Eskimos, in the context of a discussion of the arrival of the later arriving groups.
Similarly, in some contexts, a person of Bantu ethnicity in Southern Africa might be called indigenous or autochthonous (e.g. relative to Europeans and migrants from China and India) even though in other contexts that term might be reserved to the Khoisan (a.k.a. Bushman) people who resided there until the Bantu migration reached that part of Africa ca. 1000 BCE.
Some of the distaste for the use of the word "Aboriginal" comes from the historical extreme level of prejudice that was suffered by Aboriginal Australians which Native Americans and others do not with to be associated with and which carries a lot of historical baggage.
It can be compared to the terms "Idiot" or "Moron" which at one point in term were technical terms to refer to individuals in particular low IQ ranges but developed a derogatory sense and as a result came to be disfavored.
Prior to "Native American" the common term to refer to indigenous people of the United States was "Indian" (and it remains the correct term of art in U.S. law relating to Native Americans).
But, in addition to being ambiguous (because the term "Indian" means both person from India and Native American), it was also developing a moderate derogatory sense and was increasingly not a term that Native Americans felt comfortable applying to themselves. The term "indigenous" or "aboriginal" was never in wide use to refer to Native Americans outside of anthropology circles.
In part, this happened because the concept of "indigenous peoples" was not well developed at the time of first contact with Columbus and early European settlers of the Americas when the indigenous people of the America were Christened "Indians", but by the time that England and Aboriginal Australians was initiated, the more general concept of an "indigenous" or "Aboriginal" people was better established.
Unlike, Inuit, however, "Native American" was never a self-referential term of these peoples for themselves and unlike Inuit it is a grouping of a variety of peoples rather than to a single coherent ethnicity and identity. Generally, a Native American is a Sioux or Apache or Navajo or Ute first, and a Native American second.