A caliphate is a historical Sunni Muslim institution of leadership that combined political and religious offices. The first caliph was Abu Bakr (successor to the Prophet Mohamed) in the seventh century and the last caliph was Abdul Mecid II, the Ottoman Sultan deposed in 1924. As with most religious terminology, the "caliphate" is a contested term, but a leader or caliph of most Muslims existed for thirteen centuries in one form or another. This often became dynastic power in the monarchies of the the Umayyads (661-750), Abbasids (750-1258), Mamlukes (1250-1517), Seljuks (1038-1194), and Ottomans (1300-1924).
Shia Muslims do not believe in the legitimacy of the Sunni caliphate, arguing for an imamate that is drawn from the family and bloodline of the Prophet Mohamed.
In theory, the caliphate is expected to enforce sharia as state law and maintain unity of Muslims under one ruler. In practice, different interpretations of sharia have been adopted by assorted caliphs, and from the earliest of times, there was more than one caliph. For example, the Umayyad dynasty was based in Damascus in the eighth century, but there were rival claimants to the Umayyad caliphate in Andalusia and Cordoba.
Radical Islamist groups from the non-violent Hizb ut-Tahrir to the violent Al-Qaeda or ISIS all seek to re-establish a caliphate that applies their understanding of sharia as state law. This more literalist version of the sharia advocates for Jihad as a means to creating the caliphate, and as a way of expanding the territory of the state. It is an expansionist state with no fixed borders, and non-Muslims who come under its rule are expected to pay special tax (jizyah) for protection. Muslims in such a caliphate would be judged by extreme readings of the sharia which include punishment for adultery by stoning, or amputating the hand of a thief, a lack of religious freedom or women's testimony in court being valued less than that of a man.