Historically, a caliphate was an area where religious Muslims lived under the leadership of a caliph, who was considered to be a successor to Muhammad, Islam's prophet and founder.
A caliphate is a kind of state, but because it's based on religion, it's not bound by the international laws that govern nation-states. The earliest caliphates were formed after Muhammad's death in 632, as a way to continue his teachings. The last political caliphate, in the Ottoman Empire, ended in 1923. In more recent times, the extremist militants known as ISIL have declared themselves to be a caliphate. The word is rooted in the Arabic khalifa, "successor."