Etymology
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American (n.)
1570s, originally "one of the aboriginal peoples discovered in the Western Hemisphere by Europeans," from Modern Latin Americanus, from America (q.v.). The original sense is now Native Americans; the sense of "resident of North America of European (originally British) descent" is from 1765.
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American (adj.)

1590s, "pertaining to the Western Hemisphere and its aboriginal inhabitants," from Modern Latin Americanus, from America (q.v.); the sense of "pertaining to the residents of North America of European (originally British) descent" is first recorded 1640s; later "pertaining to the United States." French Américain, Spanish and Italian Americano, German Amerikanisch. Fem. form Americaness attested from 1838. The American beauty rose so called from 1886. American English as a sub-language attested from 1806; Amerenglish is from 1974.

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all-American (n.)
1888, plural, as the name of a barnstorming baseball team composed of players from various teams across the United States. From all + American.
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un-American (adj.)

"not characteristic of American principles or methods, foreign to U.S. customs," 1818, from un- (1) "not" + American (adj.).

Everything is un-American that tends either to government by a plutocracy or government by a mob. [Theodore Roosevelt, 1917]
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Anglo-American (n.)
"English person who has settled in North America," 1738, from Anglo- + American. Originally often in contrast to German immigrants; later (1830s) in contrast to French-Canadians, Louisiana French, Spanish Mexicans. As an adjective from 1797, "pertaining to the English who have settled in America;" meaning "pertaining to both England and the United States" is from 1812.
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African-American (adj.)
there are isolated instances from the late 18th and early 19th centuries, but the modern use is a re-invention first attested 1969 (in reference to the African-American Teachers Association) which became the preferred term in some circles for "U.S. black" (noun or adjective) by the late 1980s. See African + American. Mencken, 1921, reports Aframerican "is now very commonly used in the Negro press." Afro-American is attested in 1853, in freemen's publications in Canada. Africo-American (1817 as a noun, 1826 as an adjective) was common in abolitionist and colonization society writings.
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Anti-American (adj.)

also antiamerican, 1773, in reference to British parliamentary policies, from anti- + American. As a noun by 1788. Related: Anti-Americanism "opposition to what is distinctly American" (1844).

The term "anti-American" is a loose one, and loosely employed. My own working definition of it, admittedly a slack one also, is that a person is anti-American if he or she is consistently contemptuous of American culture and furthermore supports any opponent of U.S. policy, whoever this may be. [Christopher Hitchens, review of "The Life of Graham Greene, Vol. II," The Atlantic, March 2005]
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American dream 

coined 1931 by James Truslow Adams, U.S. writer and popular historian (unrelated to the Massachusetts Adamses), in "Epic of America."

[The American Dream is] that dream of a land in which life should be better and richer and fuller for everyone, with opportunity for each according to ability or achievement. It is a difficult dream for the European upper classes to interpret adequately, and too many of us ourselves have grown weary and mistrustful of it. It is not a dream of motor cars and high wages merely, but a dream of social order in which each man and each woman shall be able to attain to the fullest stature of which they are innately capable, and be recognized by others for what they are, regardless of the fortuitous circumstances of birth or position. [Adams]

Others have used the term as they will.

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Amtrak 
U.S. government-run railway corporation, 1971, contraction of American Track.
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Americanize (v.)
"render American in character," 1797, from American + -ize. Related: Americanized; Americanizing.
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