The type of government that allows their people or citizens to vote for their leaders is system and process called democracy. There are two types of democracy, indirect and direct. Direct democracy enables people to vote and elect straight to the laws, rules and decrees regardless of elected individuals as a matter of fact. Indirect democracy is where the vote and elect certain representatives. And these representatives choose which law or decree can be passed. Hence, you maybe referring to indirect democracy.
Currently there are 123 democratic countries.