The Holy See (or Vatican City) in the heart of Rome is perhaps the modern world's only pure theocracy. Treaties with Italy established the Vatican as an independent state, headed by the pope, spiritual leader of the world's Roman Catholics.
Read more about types of government ruled by a religious leader in the related link.
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The word "theocratic" comes from the Greek words for "god" and "government. It means "government by priests" or whatever kind of religious leaders the particular state has. A "theocratic republic" would be a state whose governors are elected from a group of priests. Vatican City is a "theocratic republic". The government of Iran, although described as a republic, is really more like a theocratic constitutional monarchy, as the Supreme Leader (Ayatollah) is not an elected position (but is one which is always filled by a top cleric.)
An organized government is one that has leaders and structure. It might have a single leader, or numerous leaders with different levels of authority. An organized government would have departments, branches, and levels of control.
A theocratic government is one where the Religion and the State are the same - that is, all government functions (and decision-making) is done by the religious hierarchy. This is distinct from the concept of a State Religion. Currently, there are really only one real theocracy in the World: Vatican City. Iran has a strange mix of theocratic and democratic elements. Likewise, Saudi Arabia has a mix of theocracy and an absolute monarchy. Israel has a State Religion with some influence over the secular government, but even that portion really isn't a theocracy (as the State Religion isn't organized in such a manner as to be able to exert institutional influence).
That will depend on the country, as different countries have different titles for the head of the government. Prime Minister is a common title, but there are many others although they have a similar role. It is also possible to have more than one party in a government, so in that case only one of the leaders will be the head of the government and the other leader or leaders will have different titles.
In the past, most governments had similar charecteristics, that there was one ruler with almost ultimate power. Some variations occur, however, in how rulers gained their power. Most early governments were a theocracy (religous figure rules), but eventually political leaders became more militarily powerful. However, as history goes on there are more checks and balances on governmental leaders. The first democracy, which was in Athens, Greece, was a first step towards self government. This tradition was continued on in Rome until Octavian made Rome a dictatorship. After this, we see feudalism, with many strong nobles taking power and then large Monarchies with one ruler until the signing of Magna Carta in 1215, which took power away from kings. Self rule is again seen in formation of a Democratic Republic in the United States.