Use of these terms has been practiced and advocated by some non-American people to distinguish U.S. nationals from people living in other countries in the Americas. In practice, this is not usually necessary in English because American without any modifier (like South American) is well-understood to be a U.S. national and nobody else. In other languages, notably Spanish, American is sometimes more ambiguous.
The concern on the part of advocates of these terms that motivates use of the word is that, since America is part of the names of both North America and South America, it follows in their logic that American ought to be understood to mean, "inhabitant of the Americas".
Some people who deprecate use of "American" often do so as part of criticism of the people or the government of the United States, especially criticisms of US foreign policy towards the rest of the Americas. Such criticism has focused on perceived or stated policies such as Manifest Destiny and the Monroe Doctrine.
Since various people have found American to be ambigious since the beginnings of the U.S., there are many variations of this idea and term. Other words that have been suggested for the same purpose are Columbian, Columbard, Fredonian, Frede, Unisian, United Statesian, Colonican, Appalacian, Washingtonian, Usonian ("Usonian" is an adaptation from Esperanto, apparently coined by Zamenhof), Uessian, U-S-ian, and Uesican (in approximately historical order from 1789 to 1939, according to Merriam Webster's Dictionary of English Usage).
Other examples observed in the field: