American simply refers to the indigenous peoples of North America or South America, or to a citizen of the U.S, or is generally a reference to the United States.
An empire is a major political unit having a territory of great extent or a number of territories or peoples under a single sovereign authority, or it is something resembling a political empire or an extensive territory or enterprise under single domination or control
If we put these together we get American Empire. I don't think the term is meant to be offensive, and I don't think, as cathcon suggests, that it is meant to sound cool. Is it modern? Well, to the extent that there were many American empires (the olmec, the huastec, the mixtec, the aztec, the Tihuantinsuyu, the Maya, etc.), long before the age of conquest, it is not necessarily modern. But to the extent that it specifically refers to the hegemony of the US in world affairs, then it is fairly modern. The Monroe Doctrine is probably where it began, but it has since evolved into: (1) the political unit into which the United States has evolved and the vassal states in the Americas subject thereto; (2) a general reference to the cultural, military, and economic hegemony of the United States.
I think it's a reasonable and inoffensive term. It is also useful, since it reminds of how badly twisted our government has become since the days of George Washington and Thomas Jefferson. I guess I'm a bit optimistic in the sence that such a reminder will act as a slap in the face. I'm no fan of The Donald, or of his wall, his impropriety, his general disdain for polite conversation, or of his lack of geopolitical curiosity, but I have to admit that his anti-imperialist rhetoric has some appeal. We might applaud those who remember that imperialism doesn't really best suit our economic ends.
Jefferson himself described the United States as the Empire of Liberty.
