Republic means a form of government in which the supreme power is held by the people. The elected individuals represent the citizen of the country and exercise power according to the rule under the constitution of the country.
It also means a state when sovereignty rest with the people or their representatives rather than with a monarch or emperor. It simply means a country with no monarchy, which elects its leaders.
It is also a state of government in which the executive and the legislative branches of government are separate.