The century opens with America establishing itself as an empire and a military power. From then on it was a mostly steady increase in America's economic and military influence on the world. From World War I and our diplomatic role, through our period of isolationism into World War II, where we emerged as one of only two remaining empires that were functional. Us and the Soviet Union were superpowers, that influenced the course of world economies, the nature and makeup up foreign governments and the structure of military alliances.
In the new century, America faces new challenges from an unstable resource supply, mounting debt, protracted guerrilla wars and a terror threat in the midst of economic recession and the loss of manufacturing jobs to globalization.
This could take up a book. Infact, I would suggest reading Clive James' "Fame in the 20th Century" as a good starting point. I would suggest that the American "presence" in the 20th century would be where some of its most powerful impacts have been felt. World conflicts, economic policy, military strength, fashion, culture, scientific endeavors are but a few of the many areas in which American presence has been felt in this century. I think that this "presence" has been brought into some new configuration in the 21st century, an era defined by globalization, information sharing, terrorism, and economic crisis in only the first ten years. The changing paradigms and senses of understanding would be where America lies. Such challenges must be ready to be faced in the 21st Century.
To me, historians call it the American Century because the United States was central to everything that happened in the century after WWI (and the US even had a large part in that).
It was US might that was key to deciding World War II. The competition between the US and the USSR dominated the world from the end of WWII up to around 1990. After that, the US was the only superpower for the rest of the century.
In addition, the US was the economic powerhouse of the world after WWII. The US economy boomed, and that pulled economies like that of Japan and, later, South Korea, with it.
So the US dominated the 20th century in terms of military and economic power.