The modern term is derived from the Latin imperium, which was coined in what was possibly the most famous example of this sort of political structure, the Roman Empire founded in 31 BC. The actual political concept, however, predates the Romans by several thousand years. Probably the first example was the Akkadian Empire of Sargon of Akkad.
An empire can take several forms. Empires have been traditionally ruled by powerful monarchies under the leadership of a hereditary (or in some cases, self-appointed) emperor. Historically, most empires came into being as the result of a militarily strong state conquering other states and incorporating them into a larger political union. Often these multi-ethnic entities were strengthened by the introduction of a common religion, as was the case under Constantine I of the Roman Empire.
The discovery of the New World provided an opportunity for many European states to embark upon programs of imperialism on a different model, colonization. Under this model, subject states were de jure subordinate to the imperial state, rather than de facto as in earlier empires. This led to a good deal of resentment in the client states, and therefore probably to the penultimate demise of this system in the early twentieth century.
Another problem with the European imperial model might be described as gerrymandering. In the interest of expediency, an imperial power tended to carve out a client state based solely on convenience of geography, while ignoring extreme cultural differences in the resulting area. An example of the attendant problems can be seen in India. Formerly part of the British Empire, after India gained its independence it quickly split along cultural lines producing the country of Pakistan, which later split yet again resulting in the country of Bangladesh.
The concept of "empire" in the modern world, while still present politically, is losing cohesion semantically. For example, the former Soviet Union fits many of the criteria of an empire, but nevertheless did not claim to be one, nor was it ruled by a traditional hereditary "emperor" (see Soviet Empire). In the early 21st century, tendencies to refer to the USA as an empire could be seen, following the book Empire by Antonio Negri and Michael Hardt (see American Empire or History of United States Imperialism).
Among the empires in history are: