Answer: You know that after the victory of Islamic revolution the most critical decision was made on establishing a new political system.
Different groups and parties had their own views and suggestions. Some supported a democratic system and voted for a republic, some believed in an Islamic government and others desired an Islamic republic administration.
Imam Khomeini however deemed for a type of system which could be established based on both democracy and Islamic laws. Therefore he suggested Islamic Republic as a substitution for the monarchical corrupted system.
Eventually, the new system was established through a referendum on 12 of Farvardin 1357 titled the Islamic Republic of Iran.
From this perspective, Republic stands for the form of government while Islamic expresses its content and originality.