Defining "empire" is not particularly easy. The etymology of the word dates back to Republican Rome where the title Imperator was granted to victorious generals, and the lands they had conquered were referred to as Imperium. Eventually the title gained political connotation, and leaders symbolically hailed as imperator by their legions was generally the first step of claiming legitimacy before rebelling against the current Princeps or Dominus (which we often refer to historically as the "Emperor" today).

Following the end of the Classical Period, Emperor was a title rarely claimed in Middle Age and Medieval Europe. In 800, Charlemagne was crowned Holy Roman Emperor by the Pope, which revived the title in the West. However, the HRE is a massively confusing office and political entity (if it can even be described as such) which won't help understanding an Empire at all, so we'll skip it entirely. The more or less most common attributes of an empire in the west (excepting the HRE) was when one established hereditary kingdom absorbed multiple previously independent hereditary kingdoms and forged a new singular entity. This, of course, is not always the case: when the English crown was united with the Scottish and Welsh crowns, it became the United Kingdom, not the British Empire. Yet when the German states were united under the Prussian Kingdom, it did become the Empire of Germany. The Austro-Hungarian Empire proclaimed itself an empire in 1805 following Napoleon's dissolution of the HRE, and was recognized as such by other European powers.

European colonial acquisitions were very commonly referred to as Empires (The Spanish, Portuguese, French, English, Dutch, etc.), but these did not always amount to the acquisition of a formerly independent states incorporated into the imperial country's polity, but could be colonies of settlers displacing existing native populations. The British evolution from client states to direct administration on the Indian subcontinent differs from this version as well, however. In the end, empire is difficult to clearly define because there is almost always an exception - and that's only counting the Western world. The various empires of the near and far East are a different animal altogether.

An argument can be made both ways as to whether America constitutes (or ever constituted) an empire. The acquisition of The Phillipines, Cuba and Puerto Rico following the Spanish-American War; the annexation of Hawaii; the territorial gains following the Mexican-American War all lend themselves to the argument that America was an imperial power. However, there was never any office of emperor, nor any constitutional precedent for the administration of imperial or colonial holdings. So were we an imperial power? A colonial power? Eventually, sovereignty was restored (either willingly or de facto) in Cuba and The Phillipines, and American Citizenship (limited citizenship anyway) was granted in all other American territorial holdings. It can be said that this ended the era of the American Empire, if that's even an accurate description to begin with.

The network of American military bases across the globe that has been in place since the end of WWII is sometimes referred to as the American Empire, but this is largely a politically motivated description, and doesn't hold up to scrutiny. Military bases are maintained by treaties between allied states each recognizing the other's sovereignty. American military personnel are still subject to the host country's laws, and the US does not claim the land bases are located on as sovereign territory. I'm hard pressed to find an historical example of such a situation being referred to as empire.

Following the war, colonialism and imperialism have taken on a highly politicized and controversial connotation, and cannot be compared to the concepts of empire in existence even a century ago. As such, it remains a bit subjective and divisive a description which - as I said in the beginning - is hard to accurately define.