A republic is a form of government where the power lies with an elected body of representatives representing the citizens… however "citizen" is being defined. Usually, the power of the government is clearly defined by law - though that law might be open to interpretation.
The dividing line between a republic and a democracy is broad and somewhat blurred - in common usage, republics stand in contrast to monarchies and so may have virtually any form of government, but to be sociologically correct, it is best to confine the use of the term to those states where the elective franchise is either extremely limited or somewhat indirect.
The extent of election can also vary - the Romans, for example, elected their ruling magistrates, but membership of the day to day governing body, the Senate, was unelected and mostly plutocratic with a few seats made available on the basis of merit (usually military valour). Many Greek urban republics - often called democracies - also had a governing body (typically called a boule) in which membership was an inherent (and usually compulsory1) part of citizenship but again, magistrates were often elected2.
Whiggish historians have tended to consider republicanism as a stepping stone to "more advanced" democratic government - although others have raised it as a "perfect" form between the tyranny of kings in the form of monarchy and that of "the mob" (in the form of democracy).
Game and Story Use
- The fun is in determining who gets to be "citizen" and thus be eligible to vote. Everyone above a certain age? Or only men/women/hermaphrodites? Only certain ethnicities? Those with a certain minimum amount of income or property? Those who have completed a term of military service, or succeeded at aptitude or intelligence tests? Those who possess magical powers? Who gets to vote says a lot about the character of a republic.
- Likewise, whether all votes count the same - for example, the Romans voted by class and by tribe, with the higher social classes and rural tribes having fewer members that the lower classes and the urban tribes - hence the vote of a member of the former counted for a lot more than that of one of the latter. These could be mixed and matched - you could, for example, be a poor farmer of the fifth class, only just qualifying for the vote, and yet belong to a tiny rural tribe with a few dozen members, or a rich equestrian of the second class from a teeming urban tribe.
- The presence of a republic is also likely to have a significant effect on your campaign world - "the awful cry of a kingless people" is likely not to go unheard, even in your typical fantasy monarchy. Warfare is probable.
- Within any republic, there will also be the counter tensions between devolution and centralisation of authority - in one direct lies genuine democracy (or anarchy and collapse), and the other the return of kings.