Republics practice the principle of "Representative Democracy". This means that power is held by the people but there is an acknowledgement that the people cannot be expected to vote on every single issue the country faces as this would be almost impossible.
The people will therefore vote a person to represent their interests and that person will then make decisions on their behalf. This is the way things are in most countries of the world today including the U.S. when you vote in Senators, Congressmen, Governors and Presidents.
