Oof, big question. I think that if we got right down to it, we'd find that definitions of empires are quite particular and vary across time and space. The one really essential element, I would argue, is that it a political body of the greatest size. Beyond that, we can find al kinds of examples of different kinds of "empires."

For example, in the modern period, European empires have usually exhibited a metropole-colony split, so that in order to really be thought of as an "empire," a state needs to control territory outside its essential "core" area. This territory typically was overseas, but not necessarily; more important was that the core area dominated the peripheral areas in different ways, extracting resources or tribute, controlling their trade relations, perhaps reforming the cultural and society of the peripheral areas, and even supplying settlers to mix with or displace the indigenous inhabitants. From about 1500 (or even a somewhat before), in Europe, these included empires that we'd consider mercantilist, that is, empires whose ideological foundation rested on the notion that peripheral possessions served to benefit the metropolis usually through commercial relationships that saw specie accumulate in the metropolis. There also might include the cultural and political relationships I noted above, but did not necessarily. So, the Spanish empire in the Americas, for example, provided tremendous amounts of silver to the metropolis, while it was also a zone of Catholic proselytization, often violently. The Portuguese and Dutch empires, on the other hand, were extensive in that they reached around the world, though they did not control a great deal of territory. Nevertheless, they existed to control trade routes and bring resources back to the metropolis in order to get a better balance of trade and preserve specie at home.

Of course, we could easily find examples that contradict this. The Holy Roman Empire comes immediately to mind, though it might be such an unusual entity that we could consider it truly unique among large states. I can't think of another "empire" so decentralized. We might also look at the Chinese empire in the early modern period, which expanded considerably to the west, into central Asia, largely by reproducing itself: settling agricultural communities that could reliably pay taxes. Certainly the imperial center saw this as good thing, a goal worthy of support and pursuit, but to my knowledge they did not pursue the kind of mercantilist, core-periphery split that underpinned most European empires.

So, hopefully that'll get the conversation moving. I'm sure there are many different examples we could cite, and perhaps someone will propose a more specific definition than mine, based on size alone.

Taking a decided post-structuralist tack, an empire is a political entity that either ascribes to itself or is ascribed by others, as one that has or claims hegemonic supremacy over other political entities or desires to claim equality with other political entities that call themselves empires.

The reason for this convoluted definition is we have to consider the many boundaries of past political entities that we consider empires, a definition that would need to simultaneously contain titular empires (The Korean Empire, hardly an empire by other metrics), structural empires (The Athenian Empire, as they did not call themselves this though historians do), as well as the gold standard empires of both (The Roman Empire, The Chinese Empire, who both were structural empires and called themselves as such according to the hegemonic authority definition).

Because ultimately, empire ^^edit: is about power and perception as much as it is about identity and identification, not just by those who would be in an empire, but those who would claim an entity is an empire.

And considering how loosely this word is used (is America an empire? Is Capitalism an empire?) we should be ready to feel mildly dissatisfied with its definitions.