The etymology of the word Hegemony traces its roots to the Greek word hēgemonía, which translates to leadership or rule. The concept was explored in Ancient Greece during the 5th century BCE, describing the city state of Sparta as hegemon of the Peloponnese. As leader or 'Hegemon', Spartan leadership forced neighboring city states to join its military alliance as well as extracting tribute from them.
The strength of a hegemon does not solely depend on their ability to exert brute force but includes their cultural, religious or economic influence.
Hegemony can exist at various levels from regional to global hegemony.
The Monroe Doctrine is an example of United States foreign policy being enabled by its hegemonic status over the American Continent. President James Monroe stated their intention of preventing new European colonies in the New World, to prevent any Great Powers from establishing itself near US shores.
Hegemony separates itself from an empire as it seeks to exert influence over another state via influence without direct ownership of the governance. A key implication being that a Hegemon doesn't incur the costs of it's own directives rather the subjected client state must bear the costs of following it's directives.
In the case of continuing defiance, the hegemon must eventually respond with strength or lose their credible deterrence with other subjected states.
In the case of the Monroe Doctrine, enforcing this policy almost led to Nuclear War during the Cuban Missile crisis, whereby the US would use any means necessary to prevent Cuba from becoming a launch site for missiles from the USSR.