Wikipedia defines the concept of American exceptionalism this way:
American exceptionalism refers to the theory that the United States is qualitatively different from other nations. In this view, America's exceptionalism stems from its emergence from a revolution, becoming "the first new nation", and developing a uniquely American ideology, based on liberty, egalitarianism, individualism, populism and laissez-faire. This observation can be traced to Alexis de Tocqueville, the first writer to describe the United States as "exceptional". The term "American exceptionalism" itself was first used by members of the American Communist Party in the 1920s, in reference to their belief that "thanks to its natural resources, industrial capacity, and absence of rigid class distinctions, America might for a long while avoid the crisis that must eventually befall every capitalist society."
Of course, the entry goes on in some detail, but the broad concept is pretty much all there; America is a unique nation for many reasons. Personally, I believe the most important distinction is that we are a nation founded in law -- we exist for a reason. Where other nations may exist because of ethnic or religious identity, America does not. And we aren't defined by our existence within the borders of an ancient monarchy or tribe. "I am American" means something very different from "I am French," for example.
The problem with American exceptionalism isn't that the concept exists -- in fact, it's not difficult to see the validity of the argument. The problem is that it sounds vaguely like an argument that it is not; that the US is better than other nations and entitled to take liberties in the world. It was the Bush administration and their defenders who argued that America the Exceptional had the right to invade Iraq, in defiance of international law.
As a result of this distortion of the concept as an expression of hyper-nationalism, Republicans have taken to demagoguing the phrase. The left does not believe that America is exceptional, they argue, which means that they believe America is not great. For them, it's an argument without facts, an empty buzz-phrase like "manifest destiny."
Of course, the biggest problem with Republicans wielding the idea of American greatness like a weapon is that there's absolutely no evidence that they actually believe it. Since Reagan at least, the Republican Party -- Reagan's "optimism" be damned -- has been characterized by a "can't do" attitude. America did great things once, built massive public works projects and is the first and only nation to put a person on the moon. But not anymore. Americans and American government are hopelessly incapable of even the simplest efforts, let alone grand projects. By Republican arguments, Americans are exceptional only in their incompetence and laziness -- a nation of boobs incapable even of running a school, let alone building a huge dam.
Take a google of the term "restoring America's greatness." You'll find articles about Sarah Palin and Michele Bachmann, as well as more local demagogues.
But you can't "restore greatness" to a nation that's already great. Republicans would savage an opponent who made that argument, yet it's extremely common on the right.
And do Republicans believe Americans themselves are exceptional? Clearly not. In arguing against extending unemployment benefits, Republicans basically say that Americans will not work unless they're forced to, that the average American will sit in front of the TV eating Fritos until their benefits run out. They believe that the bulk of Americans are lazy and dishonest.
And do they believe that America is defined by the concepts laid out in the Constitution, rather than a national or religious identity? Of course not. Find one Republican willing to argue that the US was not founded as a Christian nation. In short, they believe the national identity is defined by something that is not American.
And ethnic identity? The right is dominated by xenophobes worried about Hispanics and Muslims "taking over." Just yesterday I reported that the Tea Party Nation warned of an "invasion of America" by "non-European immigrants." America is now a Christian European country. There are a lot of those and the concept is not exactly exceptional.
So the next time you hear some rightwing commentator or a Republican politician expound on the importance of American exceptionalism, remember that there's no evidence that they actually believe what they're saying.