Seriously, I never even heard this word before I came to China, but it seems like the #1 trait of America as taught in Chinese schools. Since your Chinese teachers aren't going to explain proper context, allow me to shed some light on the USA's hegemonistic tendencies from our point of view:
1. While China steadfastly sticks to the principle that the leader of a sovereign nation has the right to do whatever he wants to his subjects, we do not believe in this. If a leader abuses his people, it is a moral obligation for those with the power to help the oppressed to do something about it.
2. Before World War I, the United States generally minded their own business more. We joined the war to help allies in Europe, even though it was really none of our business, and plenty of people had a problem with this, but England and France were really happy about it. The debate over USA's involvement caused us to stay out of World War II as long as possible.
3. After World War II, when the horrors were revealed, the American ideology shifted because we WISH we had HELPED earlier. Americans wish our soldiers had been in China in the 1930s fighting the Japanese. We wish our troops had gone to Europe earlier to save Jews from genocide. We wish we had been there in Nanjing to help you. From our point of view, *lack of hegemony* in the years immediately before 1942 cause incalculable damage on the world.
4. We think the framework for our country is perfect. People and leaders are imperfect, but at least we can get rid of them when they suck. Virtually every American believes in the U.S. Constitution wholeheartedly. It's the oldest in the world. Since it was written in 1787, every developed country in the world has borrowed from it. You should too. China is an ancient country and America is new? If you want to measure a country's age by the duration of it's current political system, America is the OLDEST. China's most recent constitution was written in the 1980s and your leaders rarely even pay attention to it. We have a system of checks and balances, you don't. We have a very strong court system to protect rights, Chinese courts are infantile.
So please, someone explain to me why American hegemony is so evil. Please try to do so without saying the word "oil." It's really rarely about that, and that's such an simplistic answer supplied by your Chinese teachers who really have little clue how international politics work. Americans really support interfering in foreign countries only to do GOOD.