In a republican form of government, the head of state is elected and not a hereditary monarch. Thus, this word denotes a government where no one person holds public power as a proprietary right.
As opposed to a monarchy, in which the head of state is appointed on a hereditary basis for life or at least until abdication, a democratic republic is an entity in which the head of state is elected, directly or indirectly, for a fixed tenure. Thus, India has a President who is elected and has a fixed term of office.