Short answers, pulled from the story.
An empire is a realm controlled by a monarch or other official and divided between a dominant center and subordinate peripheries, where the center holds political power over the peripheries. Michael W. Doyle defined it as effective control, whether formal or informal, of a subordinated society by an imperial society. Rein Taagepera called it any relatively large sovereign political entity whose components are not sovereign.
Josep Colomer distinguished them by noting that empires were vastly larger than states and lacked fixed or permanent boundaries, while states had fixed boundaries. Empires were a compound of diverse groups and territorial units with asymmetric links to the center and ran on multi-level, overlapping jurisdictions, while states sought monopoly, homogenization, and supreme authority over a single territory and population.
The word empire derives from the Roman concept of Imperium. The Latin word imperium comes from imperare, meaning to command, and originally referred to a magistrate's authority, usually in a military sense. As Rome expanded overseas, the term came to describe Rome's authority over its colonies and client states.
The earliest known empire appeared in southern Egypt sometime around 3200 BC. Southern Egypt was divided among three kingdoms, each centered on a powerful city, and the city of Hierapolis conquered the other two over two centuries before growing into the country of Egypt.
Lenin cancelled all earlier forms of imperialism and began its history in the 1760s, removing the essence of empire from politics and focusing on economics. He treated imperialism as the highest stage of capitalism, which meant it could not exist before 1876. Most mainstream historians, by contrast, recognize that imperialism predates European colonialism by several millennia.
Whether the United States qualifies as an empire is debated. Characterizing it as the American Empire is common, and Thomas Jefferson used the term Empire of Liberty, yet the country's founding anti-imperialist principles have prevented many from acknowledging that status. The 2021 Oxford World History of Empire argued that the United States, China, and Russia all pursue imperial policies despite their anti-imperialist rhetoric.