Republic is a very contested word. Even today, many Americans will insist the US was supposed to be 'a republic, not a democracy' without understanding how those words were meant and used at the time (and the fact that the distinction between them set forth in the Federalist Papers was itself a novel, and not uncontested, one).

Republic comes from the Latin 'res publica' or 'public thing', perhaps better translated as 'public entity'. Republic in this sense meant a state or political community that belongs to its citizens in common - as opposed to a kingdom which was in a sense the personal domain of the king. The leaders of a republic held 'imperium' (public authority over citizens) for the sake of the public good. This was contrasted with the rulers of a kingdom, who held 'dominium' (the private power of a feudal lord over serfs) for the sake of their own wealth and power.

A republic - a state belonging to the citizens and orientated towards the good ruled its citizens by law; the relationship between citizens and the state was not based on personal subordination or fealty, but but the laws and institutions of citizenship. The rulers of a republic derived their authority from the law, and the law form some notion of public consent. At its heart was a radical equality of status as fellow-citizens (although of course various hierarchies of wealth and status remained).

In a kingdom, in contrast, the rule of the king over subjects was profoundly unequal, and sustained by personal fealty. The law was an expression of the sovereign's power, not the other way around. At its heart was a profound inequality of status between rulers and ruled.

Crucially - and confusingly - many countries which had hereditary (or quasi-hereditary) singe rulers as their chief executive nevertheless though to themselves as 'republics'. They concentrated executive power and leadership in one person, but still asserted the notion that the state was a 'public thing', governed by law, and founded on some notion of consent.

Thus Sir Thomas Smyth's description of England under Elizabeth I was titled 'De Republica Anglorum' - although the country was clearly ruled by a monarch, he set out to show that the monarchy was a lawful and parliamentary one, founded in notions of the public interest and public consent. It was a republic with a monarchical executive, subject to law.

This had a long tradition. Even after the Roman Republic had 'fallen' and turned into the Empire, it still styled itself as a Republic. The Emperor held various titles and offices that concentrated powers in his hands, but still maintained that these were public offices, engaged in public service. The Emperor, at least during the 'Principate' period', maintained the republican magistracies and Senate and sought to present himself as the 'first citizen' of a Republic. As time went on, the pretence became less and less convincing, until later Emperors started styling themselves 'Dominus' (lord, master) rather than 'Princeps' (first citizen).

It has been argued by Anthony Kaldellis that this republican tradition continued in the Byzantine (East Roman) Empire - that the Emperor didn't 'own' the state, he was merely its first citizen, and that he had to rule with a sense of public responsibility for is actions - this was enforced by the Senate, Army and People, who together could make and depose Emperors. An Emperor who met the crowds and was faced with cries of 'Anaxios' ('unworthy') could be deposed by the Senate or Army; a provincial governor or army general who was met by shouts of 'Axios' ('worthy') could be elevated to the throne. The monarchy in that sense existed only at the service of a state that was a 'public thing'.

So it was in France. Napoleon had himself proclaimed Emperor technically by a constitutional amendment, backed by public consent in a referendum, to the republican constitution of the Year VIII. There is some discussion here as to whether that was a genuinely free and fair vote, but it seems at least true to say that Napoleon did enjoy broad popularity and that in some way his becoming Emperor represented an expression of public consent.

References -

On republican thought in general:

Skinner, Q. (1998) 'Liberty before Liberalism'

Pettit, P. (1997) 'Republicanism: A Theory of Freedom and Government'.

On the Roman Republic and Empire:

Greenidge, A. H. J. (1901) 'Roman Public Life'

Adcock, F. E. (1959) 'Roman Political Ideas and Practice'

On republican principles in the Byzantine Empire:

Kaldellis, A. (2015) 'The Byzantine Republic: People and Power in New Rome Hardcover'

On Napoleon's Republic-to-Empire transition:

Cobbin, A (1965) 'A History of Modern France, Vol. 2: 1799-1871.

Cronin, V. (1995) 'Napoleon'.

When Napoleon was declared Emperor it followed an actual election. The 1804 French Constitutional Referendum was held to determine whether to change Napoleon's title from "First Consul" to "Emperor of the French." The French public voted (allegedly) nearly unanimously in favor of changing Napoleon's title.

Now the title "Emperor of the French" itself reflects careful planning by Napoleon. Prior to the French Republic the Kingdom of France was ruled by the "King of France." The title denoted that the King held all of France and its subjects as his personal property. By the time of the Revolution this sort of thinking, that a monarch owned everyone and everything in his realm, was severely outdated, and was a main cause of the French Revolution, as people detested the idea of a single ruler owning everything and everyone.

Napoleon faced the dilemma wherein he was attempting to crown himself the monarch of a people who had just overthrown a monarchy. In order to avoid facing the same stigma the King of France did, Napoleon made his title "Emperor of the French". This differed from the King of France in that Napoleon's title did not imply he owned all of France, or that he owned the French people. It was more an implication that he was the leader of the French people, elected by them to rule. Whereas "King of France" implied the King owned France, "Emperor of the French" implied that the Emperor was merely the ruler of the French people, not the personal property owner of the entire French realm.

The subtle changing of titles to insist on a non-monarchical status by a de facto monarch has been used by various monarchs throughout history. Augustus Caesar, widely acknowledged as the first Roman Emperor, never directly called himself Emperor, instead using the term "Princeps civitatis", or "First Citizen", to show that he was not the monarch of Rome but merely the first among equals, that he was the leader of the people but was not above the rest of the people. He did this because the Romans had a history of detesting and killing attempted monarchs, including Augustus's adopted father Julius Caesar. To avoid the stigma of being a monarch while being a de facto monarch, Augustus instead continued to call the Roman Empire a "republic" and used the title of "First Citizen" rather than "Emperor".

King William I of Prussia also faced this problem when he was attempting to have himself crowned Emperor of the first German Empire. The title of "German Emperor" was carefully selected by the leading Prussian politician, Otto von Bismarck, so that it wouldn't offend any of the newly submissive German princes. They rejected the title of "Emperor of the Germans" because William, unlike Napoleon, wanted to portray himself as a ruler appointed by divine right, not by popular sovereignty. Similarly Bismarck rejected the title "Emperor of Germany" because it implied that William owned all of Germany personally, which would have offended the German Princes, and it would have laid claim to territories outside of William's realm.

Title changes have been used throughout history to make de facto monarchies compatible with de jure Republics. It wasn't so much the idea of a Republic was different back then and thus was compatible with a monarchy, it was more that the monarchs of these realms wanted to call themselves a "Republic" for various reasons, either to avoid stigmas against monarchs, to appease the populace and/or aristocrats, or to claim legitimacy through popular sovereignty. Monarchical rulers of "Republics" who wanted to avoid being seen as a monarch simply changed their titles to be compatible with Republics. Napoleon called himself Emperor of the French, Augustus Caesar called himself Princips Civitatis, and William I of Prussia called himself German Emperor. Republics weren't different in the past and thus were compatible with monarchies, rather the monarchs claimed to not be monarchs at all and instead elected leaders of the republic.