In a short work titled American Exceptionalism sociologist Charles Murray summarizes the characteristics which he thinks make, or made, America exceptional among nations.
He also gives a chapter to the criticisms that have been levelled at America, but here I'd like to focus on what he, as well as foreign observers like Alexis de Tocqueville, believes made this country special, and why millions of immigrants are confirming this judgment by their willingness to risk their lives in an arduous trek to get here.
Murray lists four elements that he identifies as "The elements of an exceptional culture:" Industriousness, Egalitarianism, Religiosity, and Community Life. He identifies three characteristics of each of these and makes the case that regarding these traits America and Americans were unique among nations in the 18th century.
Here's a short summary of his theme:
Industriousness (Self-reliance, hard work, getting ahead): It (industriousness) signifies the bone deep American assumption that life is to be spent getting ahead through hard work and thereby making a better life for oneself and one's children. The industriousness of Americans fascinated the rest of the world. No other American quality was so consistently seen as exceptional.
Egalitarianism (No aristocracy, identification with middle class, equality of human dignity): America was by no means a classless society, but Americans retained a strong sense that whatever class you were in was a matter of what you made of yourself, not who your family was, and an equally strong sense that richer does not mean more virtuous or of greater human worth.
Religiosity (Source of social activism, source of utopian aspirations, source of inner self-government): America was and still is the most religious country in the Western world and its religiosity has profoundly shaped it.
To quote John Adams, "We have no government armed with power capable of contending with human passions unbridled by morality and religion. Avarice, ambition, revenge, or gallantry, would break the strongest cords of our Constitution as a whale goes through a net. Our Constitution was made for a moral and religious people. It is wholly inadequate to the government of any other."
Community Life (Civic engagement, neighborliness, social trust): Murray uses the phrase "Community Life" to, in his words, "capture the odd combination of volunteerism, widespread charitable giving, and engagement in the community that was at the heart of American exceptionalism in the nineteenth century."
There may be other traits that characterized America and set the fledgling nation apart from most of the rest of the world in the late eighteenth and nineteenth centuries, but these four, Murray believes, are certainly among the most prominent.
Sadly, to the extent that any of them have eroded so has our uniqueness.