Hegemony is a system in which an individual or group has strong, unofficial leadership over everyone else, with the consent of weaker parties. The clearest examples of this can be seen in the ancient steppe tribes that attacked China.
In international relations, a hegemon is a state that is able to project its power and force across a given region. In many cases, a hegemonic order can be more stable and efficient, as only a hegemon can provide some collective goods, such as security on the high seas. On a world level, the United States is the only state (as of 2016) that can provide those goods. However, no state has yet been able to achieve "true" hegemony across the whole world. (We have to await "making America great again".)
Regional hegemons are more common. China was the longstanding hegemon of East Asia until the 19th century, for instance. Rome functioned as the hegemon of the Western world in the latter period of the ancient world.
The concept of cultural hegemony includes the significance language has in creating a hegemon, through a lingua franca.
 Gramsci and hegemony
Italian communist intellectual Antonio Gramsci used the term "hegemony" to describe the ways in which a society regulated what fell into the realm of the possible, the expressable or the thinkable. Gramscian hegemony is a dense and heavily-debated concept that feminist, post-colonial and post-modern intellectuals have employed in critiques of patriarchal social structures, colonialism and neo-colonialism, capitalism and neoliberalism .