The only single word or concept that comes to mind when I think of the United States is diversity. From our inception, we have been a mixture of peoples, religions and ideas, and that has continued all the way to the present day. Not only does that make us diverse, it also makes us ever changing. Take countries like Switzerland or Korea, which are pretty homogeneous and slow to change demographically, versus the United States, whose peoples and ideologies blend together in a much shorter time.
To me, as to many Europeans, The United States mean a complex of many different things, far too many to list in an enotes answer. I have visited the country twice and am in regular contact with friends and relations there, so my personal connections are very warm since I encountered nothing but goodwill and welcome, particularly on my first visit which took in the Midwest and the West Coast. On my way home I spent some time in New York which at that time had a reputation for rudeness but I enountered the very opposite.
I think what frustrates Europeans who know and like America is the paradoxical face that it presents to the world. On the one hand there is wonderful cinema, music, literature, the arts generally - much more vibrant than European, in my opinion - while on the other hand there is a sense that religious dogmatism dominates public life to a degree that we see almost as equivalent to some of the world's fundamentalist regimes. We wonder why 'liberal' is a dirty word in American politics, why Cuba is still being treated as a pariah 50 years on, why...oh, so many questions of a foreign policy nature. And yet, America still represents optimism and hope that the present crises can be overcome, that Obama will bring a new vision to a troubled and dangerous world. And of course The United States mean the greatest placenames in the world, poetry and resonance all around you as you travel through the country.