The distinction between a republican empire and a more traditional monarchy is that the people are assumed to be the source of government power.
In France, for example, voters were asked in a referendum in 1804 whether they supported the creation of an empire.
The king was later overthrown and replaced with a Republican system that protected certain rights of the citizens.
Over time, the republic broke down as the Roman senate started to cede more and more authority to the consul.
When Augustus took power the senate gave him the title of "Imperator", or "emperor" in English.