An Islamic republic is a compromise between nationalism, which is strictly and officially rejected and disdained in Islam, and the desire to impose some form of shariah (Islamic law). A more religiously correct goal would be the re-establishment of an Islamic caliphate, that is, a government of all in Islam, to deal with unbelievers in a uniform way, inside and outside states.
While many nations in the Muslim World have declared themselves to "be" Islamic republics, indeed it is almost mandatory in some regions, it is generally agreed that only Iran and Saudi Arabia have implemented such a program in any depth - and that with the unique Shia and Wahabist, respectively, strains of Islam that they alone among Muslim nations follow.
Islamic parties have been established in every democracy in majority Muslim nations, and militant Islamists have pushed this program even against oppressive dictatorships. These are often called Islamist (there is no such thing as Islamism, what the Islamists seek is Islam itself implemented as a political program, which it clearly is, as a faith with worldly requirements on its believers that include their political processes.)
Due to lack of attention to the actual provisions in Islam for self-governance, there are profound criticisms of the way in which Islamic republican goals have been expressed - many see the policies of such governments as reactionary in the extreme, and this includes many moderate Muslims. See the Wikipedia article on Islam as a political movement for a more complete history and balanced account of Political Islam as such.