The concept of republic is counterpoised to monarchy, instead of hereditary power, the power is achieved through the vote - in a direct or indirect manner. The Head of State can be a person or a group of people, nevertheless, it's worth mentioning that the concept of republic has changed over time. First of all, the Republic, in ancient Rome, was treated as an organization that excluded the kings - in other words, something of the people - in which res republica means public thing. With Cicero, the republic became counterpoised to unjust Governments, because it was the only way for a community to assert its justice. The Cicero's concept arrived into the Middle Ages, when the Republics were small and christian, having two powers: the Church and the Empire. In the Modern Age, the republic secularization happens, but the cicerian concept remains. With the monarchy and the aristocracy, the Republic was counterpoised to anarchy or violent regimes. In the Modern Age the constitution became a fundamental part of a republic, achieving an ideal of Cicero's practical reason.
Still in the Modern Age, with Maquiavelli and later with Montesquieu, a trinity of government forms was established: Monarchy, Republic (aristocracy and democracy) and Despotism - and in this typology a qualitative criterion was used, with several factors. According to these factors, a Republic must be small with relative equality, with the source of laws and social integration that must come through the virtue of the citizens who need to put the State above self-interest. Basically, in the Republic, the political order is born from below. It's worth pointing out, finally, that there are authoritarian regimes that are more republican in theory than in practice.
