We use the word “Republic” for a number of things in modern American English — to designate a kind of political system (i.e., a Republican form of government), to designate a U.S. political party (vi&, the Republican Party) — we even use it in our collective mythologies to positively describe fictional periods of bliss (e.g., Star Wars, where the good Republic gives way to the evil Empire).
The English word comes from two Latin words: Res publica. It might be rendered into English as “a thing of (or belonging to) the people”, but even with this clunky phrase, the meaning of the Latin “does not translate easily”.  Republic has nonetheless been used to translate Plato’s Ή Πολιτεία (“The Polity”), so that this work is known to us as The Republic. We seem comfortable using the term as though we were familiar with it, even if our familiarity conceals historical distance. So what does it mean, at least in the writings of Cicero? Dean Hammer reflects:
[In his work, Cicero’s Social and Political Thought, Neal Wood] has argued, for example, that Cicero refers to the state as the res of the people because the Romans “had no abstract notion of the state, thinking of it as the collectivity of citizens in much the same way as the ancient Greeks did with their polis.” But we should not necessarily assume that Cicero’s language points to a theoretical deficiency; rather, we might consider it a conceptual difference. The term “state” directs our focus to institutional and procedural aspects of how a territory is controlled; res publica, to an associational conception in which the community, though having institutional aspects, belongs to the people and is bound by affective ties organized by tradition. […] [p.31] Cicero calls a res publica a res populi, a thing or property of the people. He sometimes uses res publica interchangeably with civitas, a term that derives from the word civis, or citizen, referring to the organization of citizens into a political body. 
And in a footnote, Hammer helpfully rejects another English word used to translate res publica: regime.
The translation of res publica as “regime” is not sufficient either since it does not properly emphasize the critical role of the people in constituting the community. […] A regime refers to the organization and administration of a territory, without regard to the role or nature of citizenship. A tyranny can be a regime; it cannot, for Cicero, be a res publica or a civitas […]. 
This connects with what can be seen in Augustine regarding his understanding of the mixed polity of the saeculum.
This atomistic conception is not ours. In our day, even those who champion the rights of the individual very quickly attempt to lay hold of the machinery of the state in order to effect those rights; we are far from this Roman Ciceronian sense of what is common, which likely requires a level of homogeneity that we simply do not have.
It is fascinating how difficult it is for us to suspend ourselves when looking at another time and place. There are so many layers to the foundation of the building we inhabit, that we cannot see the bottom; most of us are not aware of what is beneath the carpet.
 Dean Hammer, Roman Political Thought (West Nyack, NY: Cambridge University Press, 2014), 30
 Hammer, Roman Political Thought, 30-31
 Hammer, Roman Political Thought, 30, fn.14