Salaam o 'alykum warahmatulahi wabarakatuh.
Muslims do believe in a leader who will be a ruler over the muslims, the whole islamic state has to live under his leadership (as long as he is sticking to the shariah' law; from the qur'an and sunnah.)
The word Khalifah refers to the successor or representative of Prophet Muhammad (sal Allaahu alayhi waSalam) or to one of his successors. This person acts as the head of state for the Muslim Ummah. Another title for the Khalifah (English Caliph) is Amir Al-Mu'mineen which means 'the leaders of the believers'.
The immediate Caliphs were
Abu Bakr As-Siddiq,
Umar Ibn Al-Khattab,
Uthman Ibn 'Affan, and
'Ali Ibn Abi Talib. These were given the nickname of Al-Khulafa'Ar-Rashidun (The Guided Caliphs).
Another use for this word is used for humanity in general. The human being is considered the Khalifah (representative) of Allah on earth according to Allah.
He will be the only ruler over the whole muslim nation, and his duty is to support the muslim nation by enjoining what is good and forbidding what is evil within the muslim nation (obviously this is every other muslims duty too.)
Non muslims are allowed to live within that nation, but it cannot be controlled by them.
They have two options:
- To become muslims.
- Pay jizyah (a small tax, because they will be protected under the islamic law, and the same way muslims pay zakaah, the non-muslims will have to pay jizyah.)
The non-muslims dont have to fight either, its the muslim mujahideen (muslim fighters) duty to fight if any disbelievers come into the land, and the people who pay jizyah are protected, unless they stop paying jizyah on purpose - then they will be fought against too, the same way muslims would be punished if they never payed zakah on purpose (without a valid excuse.)
However for the non muslims in that country - they would have to stay within the limits eg. they may not prayer the 5 prayers which are compulsary for a muslim, but they would be obliged to follow the laws such as no alcohol etc. - this is because laws such as these affect the people as a whole; eg. if a person had consumed alcohol (even if it was a non muslim within that country) - they may be in a state of intoxication and they could take any action without control eg. even something as horrendous as rape.. this would affect the society as a whole, this could lead gaps within the laws and make the society as a whole corrupt.
So you have to realise that muslims do live under a political system which is ruled by the shariah' law (derived from the qur'an and sunnah.) Even non muslims are allowed to live there, but they cannot transgress the limits, if they do - then the punishments/penalties apply to them too.
wa Allaahu a'lam. (and Allaah Almighty knows best.)
wa Salaam o 'alykum warahmatulahi wabarakatuh.