A government has five functions as listed:
Legislating Laws
Judging disputes
Enforcing Laws
Collecting and distributing taxes
Advancing a defense and military agenda
A theocracy is a government where all five are in the hands of a person "appointed by God". An absolute monarchy is where all five are in the hands of a hereditary monarchy. A representative democracy is where all are in the hands of representatives elected by the people.
Islam commands that 1 must be in the hands of Allah. And the Islamic scholars are in charge of interpreting the Law of Allah. No amount of popular opinion can change that bribery is a crime and that interest is prohibited. It is also commanded that 2 be in the hand of judges who are scholars of Islam and only judge based on the Law of Allah. (Quran 5:44–50) All five however have guidelines given by Allah that must be followed.
The rest (3, 4, and 5) are in the hands of the Caliph. The Caliph is advised and often appointed by the Shura (or advisory council). The Caliph must do his duties with the consent of this Shura. (Quran 42:38) The method of appointing the Caliph and the Shura is left ambiguous and thus can be any amount of different things. It is better for the Caliph and the Shura to be scholars of Islam and to be with the consent of the people. (Sahih Muslim)
Open rebellion and disobedience against a Caliph is not allowed though, as long as he is doing the minimums of his job. (Sahih Muslim) Non-violent protests against an unjust Caliph or ruler is always encouraged. (Sunan Ibn Majah)
In conclusion, the Islamic system is not any of the mentioned. It is not a theocracy because none of the rulers are "appointed by God" (in the sense that the Pope claims to be appointed by God) although they are supposed to maintain Allah's Law on earth. Perhaps a better way of describing it would be ecclesiocracy, but that is still a bit inaccurate because many functions of the government (especially 3, 4, and 5) can be carried out by non-scholars although it is recommended that they be by scholars. Thus there is a lot of leeway in how the government can be run.
