Under the rule of an Islamic Republic the laws of the state are the religous laws of Islam, as dictated by the Koran. The head of state and sole ruler is a high-ranking religous cleric, who rules by divine right with a group of religous leaders and spirtual advisors.
Though there have historically been many regimes that could be interpereted as primitive Islamic Republics, in the modern era there have been very few.
The Islamic Republic of Iran (1979- ) was one of the first contemporary nations to formally atempt to follow this form of a strict, Islamic government. However, many believe that since the death of the republic's founder Ayatollah Khomeini Iran's traditional religous infastructure is crumbling, and the nation is slowly edging away from it's strict Islamic principles.
The Islamic Emirate of Afghanistan (1996-2001) was also considered to be an Islamic Republic, though its regime operated quite differently than the one in Iran.
Today, the creation of Islamic Republics is the rallying cry for many hard-line Muslim fundamentalists all over the world. Many of these Muslims advocate the abolishment of the monarchies of the Middle East; regimes which they believe to be overly secular or otherwise destructive to Islam.