A republic is a form of government where the country is considered a "public matter", not the private concern or property of the rulers. Power is held by the people and their elected representatives, and there is an elected or nominated president rather than a monarch. A monarchy is a form of government in which a person, the monarch, is head of state for life or until abdication. The political power rests with the monarch, although the extent of their power can vary from absolute to purely symbolic.
Key differences:
In summary: The main distinction lies in how the head of state is chosen and where sovereignty ultimately resides.