A 'republic' is possibly best described today as a government not directly ruled by a monarch.
In monarchies, an unelected monarch controls the government, with power being passed down to the monarch's heir, or whoever they choose as their successor.
In a republic, power rests theoretically with the public. Although traditionally, a republic meant that people would elect their representatives into power, over time, the word republic has been used by governments that offer no such power to their citizens. North Korea is a particular example: it calls itself a republic, but functions more or less exactly as a monarchy.
The word 'republic' comes from the Latin 'res publica', meaning 'the public-thing'.