This question looks like it ought to have options.
You are probably looking for either:-
Constitutional Republics or Parliamentary Democracies both of which allow the people to elect representatives to form governments. They work slightly differently. There are probably also other systems.
The catch all is "Representative Democracy".
rasputin
a government in which the citizens elect their leaders
A/An republic is rule by representatives who are elected by the citizens
A government run by elected representatives is called a republican form of government.
The type of government in which citizens choose representatives to run the government is a representative government. This is the type of government that the United States borrowed from Greece once their independence from Great Britain was obtained.
REPRESENTATIVES WHOM ARE ELECTED BY CITIZENS TO REPRESENT THEM IN CONGRESS.
elected representatives
In a country were the citizens can elect representatives into the legislative branch of government, i.e. the parliament, there is a representative government. People take part in the making of laws through their representatives ( - indirect democracy).
a government in which the citizens elect their leaders
That would be a democracy.
republic
elected representatives