The speech of Allah (SWT) to the Prophet (SAW)also applies to the Ummah. To establish the rule of Allah, entails the establishment of the State and the appointment of a Khalifah, who governs according to Islam. It is incumbent upon the Musilms to establish a state and appoint a leader who applies Islam. The appointment of such a Khalifahis obtained through the Bay'aa.
It has been reported on the autority of Nafi that Abdullah ibn Umar said: I heard the Messenger of Allah (may peace be upon him) say:
"One who dies without having bound himself by an oath of allegiance (to an Amir) will die the death of one belonging to the days of Jahiliyyah." (Muslim)
It has been Narrated on the authority of Abdullah b Amir b al-As that he heard the Prophet (SAW) say:
"He who swears allegiance to a Khalifah should give him the pledge of his hand and the sincerity of his heart (i.e. submit to him both outwardly as well as inwardly). He should obey him to the best of his capacity. If another man comes forward (as claim ant to Khalifah) disputing his authority, they (the Muslims) should behead the latter." (Muslim)
In accordance with Islam, it is the duty of the Muslims world wide to elect a Khalifah. Such an appointment is seen as a duty (fard) similar to all other duties within Islam. The duty is seen as inevitable, and any divergence from the path is considered a grave sin, and therefore any neglect of this duty will be punished accordingly. The establishment of a Khilafah is seen as vital, because without it Islam cannot possibly be applied.
Khilafah is one of the most important issues in Islam, many
versus in Quran and many Hadiths of the Prophet ordered Muslims
to establish such a system. Ruling by Islam is the most frequent
issue discussed in Quran after the belief and creed. Therefore,
Khilafah was discussed by many Muslim scholars, the following are
the definition of some of them to Khilafah.
Ibn Khaldoon defined it as: A representation, of the one
who has the right to adopt the divine rules, aimed at
protecting the Deen and ruling the world (Dunia) with it.
http://www-personal.umich.edu/~luqman/