So, to get the linguistics out of the way: The term "Empire" derives from the Latin "Imperium", which literally means "right to command" (cf "imperative"), but took on a more extended set of meanings throughout late Republican and early Imperial Rome. Broadly speaking, it referred to the right to raise troops and use them to enforce your political will in a region - that region would then be your Imperium. It wasn't necessarily just a legal right, but also a form of potestas, political power. The Latin term which most closely corresponds to our notion of "Roman Emperor" in the early Imperial period is not "imperator", "commander", but rather princeps civitatis, or "the first citizen". But, importantly, the princeps civitatis had the right to command throughout the whole Roman Empire, and acknowledgement by the now increasingly professionalized army was an absolutely essential part of his recognition. Thus, given this military focus, do we get the term emperor.

So Rome went from being a republic into, essentially, a kind of hereditary military dictatorship that cannot readily be distinguished from a monarchy, although the emperors always considered themselves as being above mere "kings". But many would also say that Rome was an empire of sorts even before the formal transition, following the conquests subsequent to the Punic wars.

These two qualities then generally sum up the properties common to polities generally labelled as "Empires":

1. The ruler (or ruling institution) would generally claim a status above that of a mere king.

2. The polity ruled over a large, multi-ethnic area, with the definition of "large" being somewhat context-dependent; it's large and non-homogeneous enough to contain both a "core" and a "periphery".

The first property is typically reflected in the assumption of titles such as Great King or King of Kings. In the case of the Akkadian Empire, I am not sure off the top of my head if Great King (Sharru Raba) was used as early as Sargon, but he did use the titles "King of Sumer and Akkad, King of Kish", with "king of kish" for reasons somewhat beyond my understanding is taken to mean something like, "king of the world". Throughout the Babylonian and Assyrian empires that followed, these titles were retained and had additions made, such as "King of Many Lands", "King of Kings", "King of the Four Quarters [of the known world]".

In the case of China, the emergence of an empire is generally taken to begin with the king (wáng) of Qin, Ying Zheng. Having defeated and annexed neighbouring kingdoms, he created a new title, Huang Di, which can be loosely translated as Divine King, with both titles being traditionally associated with divnities. Huang and Di are the titles most commonly rendered Emperor in English, along with Tianzi, "the son of heaven", which is a title that precedes Qin Shi Huang Di (The First Qin Emperor, as Ying Zheng styled himself).

In the Eastern Roman Empire, something similar would happen with the title Basileus, which in Late Antiquity and the Early Middle Ages was applied to only two people - the Roman Emperor, and the King of Kings of Persia. This association with imperial rulership comes from the use of this title by Alexander and his successor kingdoms (and thus indirectly from its association with the King of Persia) which influenced Roman ideas of governance.

The second property tends to come along with the first, since the shoring up of a status above that of a king generally means that you have to subjugate neighbouring kingdoms. The more interesting aspect is perhaps the emergence of a core and periphery. The core generally contains the ethnic group of the ruling class and is the seat of the emperor, while the periphery contains various neighbouring groups. These are often managed in different ways - for example, the periphery might be allowed more self rule and be exempt from ordinary military service; it might be more heavily garrisoned, have a local governor from the ruling dynasty, and so on. Conversely the core tends to contain the nobility and ruling class, and is thus a potentially dangerous breeding ground for rivals, and it must be managed differently (for example, through marriage alliances or exchanges of favour).

The result tends to be cultural and economic exchange of various kinds between the core and the periphery. In many cases it can also lead to extractive and exploitative policies toward certain regions, population transfers and ethnic cleansing to manage rebellious populations, along with other horrors.

Ultimately, these effects give rise to a set of common behaviours between various large polities, which is why the term "empire" is useful in describing them. But it's certainly not an objective and value-neutral term and there are disputes about its applicability in a variety of cases.