For antiquity (or later in the case of societies that did not refer to modern terminology to qualify their form of government) the assessment of whether a state organisation is a republic is based on retrospective analysis by historians and political theorists.
For more recent systems of government, worldwide organisations with a broad political acceptance (such as the United Nations), can provide information on whether or not a sovereign state is referred to as a republic.
A maritime republic was a thalassocratic city-state during the Middle Ages in which the merchant class had considerable power.
Confederal republics are associations of sovereign states, usually having power over critical common issues such as defense and foreign policy: A crowned republic, is a form of constitutional monarchy where the monarch's role is commonly seen as largely ceremonial and where all the royal prerogatives are prescribed by custom and law in such a way that the monarch has limited discretion over governmental and constitutional issues.
Federal republics are federal states in which the administrative divisions (states or provinces) theoretically retain a degree of autonomy which is constitutionally protected, and cannot be revoked unilaterally by the national government.