Depends on the definition you use of "empire."
The strictly technical definition is any political entity whose head of state takes the title "emperor."
The classical definition is any state that rules over more than one ethnicity.
The informal usage is any state that severely exploits the resources of a certain socio-political group for the favor of another.
Well... they apply pretty well to more "recent" empires like Russian Empire or Hapsburg's Empire but if one applies such definitions to the Roman Empire he'll get quite weird results.
- "Emperor" head of state
In first the roman emperors for quite long time did rule giving more or less lip service to the republic and the senate.
Effectively the title "emperor" was used as temporary honorific for successful military generals.
"Imperator", the latin word from which "emperor" come from, was a title originally roughly equivalent to commander.
Only after the reign of Tiberius, the act of being proclaimed imperator was transformed into the act of imperial accession. In fact, if a general was acclaimed by his troops as imperator, it would be tantamount to a declaration of rebellion against the ruling emperor.
- State that rules over more than one ethnicity
The roman state used to rule over multiple ethnicities in republican times when the state was far from an empire in the classical term: rulers did not inherit the title for life but they were appointed by the senate for one year sharing the power with another person (it's like if you get two prime ministers at the same time).
- The informal usage is any state that severely exploits the resources of a certain socio-political group for the favor of another
This pretty much cover every existing state in the world, in the past, present, and (most likely) future.
In any case the roman state started assuming the form of empire after Caesar:
- one only person as head of state
- title is mostly inherited
- head of state has absolute power over all other parts of the government
- head of state is head of military (supreme commander)
The roman state under Julius Caesar did not have all these characteristics, only under Augustus the roman state got them.
Anyway, coming back to the original misconception: Augusts did not call himself emperor, but to all extent can be considered the first emperor of Rome.
Augustus ruled as "princep", meaning chief or first, a kind of "first among equals" among the citizens of Rome to avoid other titles abhorred by the romans, like the title "Rex" ("king"), and after Julius Caesar also "Dictator".
The Latin words for "emperor" were "Caesar" and "Augustus."
not completely true.