Answer:
Depends, but it's often through a period of revolution, where the state has failed to cooperate with its citizens, empowering a vocal organization to become the government.
Through violent means, such as the Russian Revolution, the Russian revolution abolished the monarchy and promoted the Bolsheviks to a position of power, which as history shows, led the nation into forming the USSR.
Augustus Caesar was victorious in a set of civil wars across the Roman Republic, and changed the form of government from a republic into a dictatorship headed by the emperor.
Or, there are peaceful revolutions, such as the 1973 Afghan revolution, which saw the shift in power from the monarchy to the people in the form of a republic.
Francisco Franco, fascist dictator of Spain, restored the power of the monarchy using his own power, and the monarchy used his restored influence to transition Spain into a democratic government.