But what is khilafah? Well, it a large topic and I will only give a summary here. Khilafah literally means - representation. It is a representation of Allah's dominion on earth.
A khalifa is a representative of Allah, ruling over the creations of Allah on earth, according to the rules set out in the Quran by Allah Himself. It is a just rule. This representation is merit based, and not birth based.
What it means is that the ruler is elected by a council (shura) on the basis of merits. One does not become a ruler on the hereditary basis, inheriting the office. Kingdoms and Monarchies are frowned upon in Islam.
"Khilafah ala minhajun nabuwwah" - Khilafah following the Path of Prophet Mohammed (pbuh), is considered the perfect model for Khilafah. A just and kind ruler, who really cares about each one of its subject.
It is to be noted that there is no separation of state and church in the Islamic system of rulership, which means the Khalifa rules by the methods prescribed in Quran and Hadeeth, guided by the religious scholars and political advisers.
As we all know, there are certain things made very clear, and certain things depend upon the current situation and ijtihad of the khalifa and his council. The aim is always the benefit of mankind and a good peaceful life for the subjects.
Since Islam is a political ideology which is beyond status, race, nationality, and border, a khilafah is for the entire Muslim ummah all over the world, working as a uniting and protecting factor. Last but not least, Wailun lil al-'Arab min Shar Qad Iqtarab.