From the Greek word ἡγεμονία (hēgemonía), meaning "authority, rule, political supremacy," which itself is derived from the noun ἡγεμών (hēgemōn), meaning "leader," and the verb ἡγέομαι (hēgéomai), meaning "to precede, lead the way, guide," a hegemony today is the political, economic, and military exertion of power and control of one state (country, nation, kingdom, etc.) over another.
In 5th century BCE Greece, Sparta possessed a hegemony over the Peloponnesian League, which included other city-states like Corinth, Kythira, Melos, and Pylos, among others. Following World War II, the two main hegemonies in the world were the Soviet Union and the United States, who both expanded their spheres of influence and fought a number of proxy wars with one another (e.g., Korea, Vietnam, Central America, etc.) during a period called the Cold War.