The current usage of the term "hegemony" usually refers to a spawn of political science called Hegemonic Stability Theory. This is an attempt to define international systems, or world orders, and the rise and fall of great powers, the assumption being that these things are systematic.
This theory discusses hegemons; the term refers to a country, not the leader of a country as in Enders Game. In fact, leaders are considered the mere midwives of hegemonic eras. A hegemon is defined as being the controller of a necessarily unequal, economic system that is fueled by the self interested cooperation of other nations. (I'll explain this) While it is true that hegemons are powerful in their own right, it is their system that allows them to manipulate world events.
As this theory goes, the first country to become hegemonic was Portugal. At the time, wealth in Europe was gained by bringing spices from the Far East back and selling them at a much, much higher value than solid gold. The Portugese galleon did this the best, and by using these to control the spice trade, they forced other countries into an economic system which was favorable to Portugal. The other countries needed to participate in order to gain wealth, self interested cooperation, and that wealth was determined by how good of friends they were with Portugal. Note that the Portugese did not need military to put down contenders; their strength was in their economy, as it was much more profitable to go with the hegemon than against it. In this way, they used profit potential and motivation, economic devices, to control an international system.
When this system changed, it was due to a change in technology. Suddenly, The Netherlands were building ships better than the Portugese. They built them faster, more efficiently, and they built more of them. Control of the spice trade shifted, and so did the hegemonic system. Countries close to The Netherlands began profiting by having good relations with them, and countries with bad relations did poorly. This is the same system that Portugal contolled, but in different hands. A hegemon naturally creates a core group of countries and benefactors which fades into a periphery as relations weaken.
In these examples it is more important to see they "why", than the "who." Portugal did not become powerful and in control because they were tough; they did so because they had a system, an economy, and a technology that worked. The same goes for the Netherlands. If you can see how these elements are working together to produce such a ruler, then you can grasp the idea of Hegemonic Stability Theory.
With the invention of a better clock, which could keep time on a boat, the British were able to calculate longitude. This was a huge seafaring advantage, and, obviously, a second shift occurred. Britain grabbed the reins of the spice trade and became the next Hegemon. This continued up until the point where the spice trade as an economy lost its potency. The major countries began to set up colonies around the globe, and France began to challenge Britain for control.
Remember in the United States' Revolutionary War, when the French sent over some soldiers to help train their soldiers? In history class I was always taught it was just because the French were really nice people. Hegemonic Stability Theory better explains their motivations because while this did add considerably to their good relations, it was far more important that Britain, France's major enemy at the time, lost a colony (US) as a direct result. So much for that mystery.
Still though, Britain's empire did well enough to maintain hegemonic control up until World War I. After this, Britain was devastated, along with the rest of Europe. For the first time, there was no world Hegemon... until, the president of the United States, an economically strong country, proposed The League of Nations. This document would have propelled the US into the status of a hegemon, had it not been voted down. Thus, until after World War II, there was no guiding force in the world. Hegemonic Stability Theorists argue that this is the cause for the rise of Fascism in Italy, Germany, and Japan. A few argue that it helped create Communism.
After World War II the U.S., for whatever reason, decided to step up to the status of hegemonic control. It did so with a very specific plan, which should emphasize and help explain the main points of this theory.
First, the U.S. knew that it would have to control the world through its economy. So, it asked all the world's nations to industrialize. It even gave them money, something capitalists simply do not do, in the name of the Marshall Plan. Industrialization causes a production of a specific good above the means of consumption for that country. This in turn forces that country to sell those excess goods for profit. This in turn creates a system of international trading. This in turn, brings the countries into the U.S.'s system, establishing it as a hegemon and making the world more stable.
To help this aim, the U.S. created many new structures, two of which being the World Bank and the IMF or International Monetary Fund. This, without going into much detail, is an organization that loans money to countries that are unable to pay it back. Whatever money these countries do have is taken, and then crippling policy instituted to take more money away later. It is interesting to note that no country that has ever borrowed from the IMF has gotten anywhere economically; in fact, they fall further into debt. Thus the contoversy between Conventional and Radical Development Theories, and Rage Against the Machine. An interesting side note here, is that responsible for all of this money is, in writing, a very specific party. If you haven't guessed, it begins with the word American and ends with Taxpayers. Ha.
There are two key points that need to be stressed about this definition of hegemony that hopefully differentiate it from simply a big word for whoever is currently in power. First, a hegemonic system is by nature inequal, and not only promotes but demands the exploitation of weaker countries by the stronger. Read: IMF. While I'm not suggesting that it was the United States' intention for the IMF to exploit in the way that it did, this theory certainly proposes that the resulting exploitation was unavoidable in the sense that it is part of any hegemonic system. Second, this inequal system is not merely prone to entropy, but destined to global war.
Because of the Nuclear Peace Theory of nuclear missiles, this has not happened yet in the current system. However, this theory points enough warning signs to bring it distaste and discredit with those not ready to believe that their world could soon be coming to an end. A signal of a declining hegemon is a militarization, followed by a collapsing economy. Sound familiar? The turning of a country from its economic to military powers is a sure sign that its economic strength, the core of its hegemonic power, is failing. Secondly, a hegemon's power is determined by the amount of cooperation it finds in other countries. Considering the foul state of the IMF, which actually ran out of money and needed to borrow imaginary money to pay to countries already in farcical debts, and the separation of policy that much of the world is taking with the United States' wartime politics, questions must be asked. Is this theory right? Is the world heading into a very dangerous time? If so, what to do?
Hegemonic Stability Theory is a very broad and complex topic, and so I am very sure that I made mistakes or left things out in my writeup. Please feel free to message me with corrections or comments.