A caliphate is, to put it simply, an Islamic monarchy. The leader of a caliphate is called a caliph. When reading about the Rashidun caliphate, you can sometimes see them being referred to as amr-ul-Mu'minin, which translates to "leader of the faithful".

A caliphate implies theological leadership of Muslims all over the world. That is why in my country (Malaysia), we have sultans and a raja, but we won't call them "caliphs" since they never made the claim of global representation (and if they did, it will be laughed at). As trivia, Saladin, even for all of his reputation, is a Sultan of the Ayubbid dynasty, not a Caliph.

The first caliphate consists of the four closest sahaba (Companions) of the Prophets - Abu Bakar, Umar al-Khattab, Ali bin Abu Talib and Uthman bin Affan. This is known as the Rashidun caliphate; unlike subsequent caliphates, all four are "elected" (bai'ah), rather than passing the "throne" to their offspring.

After a whole lot of conflict, the Rashidun caliphate ends, and the Umayyad one begins, started by Muawiyah b. Abu Sufyan. The Umayyads manage to expand their empire until it reaches Hispania (Spain).

However, the Umayyads are hostile to the Shi'ites, and this mistreatment eventually culminates to rebellion and the formation of the Abassid caliphate. However, an Umayyad prince managed to escape, and hold on to, their holdings in Spain, and established the rival Caliphate of Cordoba.

My knowledge after this gets all fuzzy so I will end this post; there are subsequent caliphates, such as the Fatimids and of course, the Ottomans.

The caliphate is a form of government with a leader that combines leadership in politics and religion. Muhammed, having no male heirs, had a lot of succesors, that widened the influence of islam. Their form of government was the caliphate. The Umayyads are a chain of successors from the same clan, if you will, not that different from european dynasties.

Aside from being the spiritual leader of Muslims, Mohammad was also the ruler of the lands that Muslims had conquered. After his death, Muslims needed a new ruler. These new rulers were caliphs which in Arabic means successor. Caliphate is the system of Islamic government that is led by a caliph.

The first four caliphs are called Rashidun which means rightly guided. These caliphs (Abu Bakr, Umar, Uthman, Ali )were chosen by prominent Muslims and chiefs of various Muslim tribes. Although there was much disagreement in choosing these caliphs, they had considerable popularity among Muslims and all of them were some of the first people who'd converted to Islam. Also, aside from Abu Bakr who only ruled for 3 years and died of old age, all of these caliphs were killed and are considered to be martyrs by the majority of Muslims.

One important point here is that Shias generally don't accept the first three caliphs and consider them to be usurpers of the position. They believe that Mohammad chose Ali as his successor and the position of the leadership of the Muslims (which is called Imamat among Shias ) belongs to him and his descendants.

After Ali was killed, his son Hassan ruled for a very short time. He abdicated and Muawiyah ibn Abu Sufyan became the first ruler of the Umayyad dynasty. He chose his son Yazid as his heir and from here on, caliphate became hereditary.

In 750 and after Abbassid revolution, they took power and called themselves caliphs. They ruled for a long time and during their time, many other kings and rulers in the Islamic world deferred to them and accepted their position as the caliphs even if they were weak.

After the Mongol invasion and sack of Baghdad, Abbassid caliphate came to its end (although it was reestablished in Cairo, they had lost most of their power) several dynasties such as Fatimid called themselves caliphate but the most well-known and lasting one of them is ottoman caliphate which lasted until 1924.

To put it in a Western context, it would be like if the President of the United States also declared himself the Pope. The Caliphs claimed not just political authority, but also religious authority as the heirs to the Prophet Mohammed. That's what sets them apart from Sultans, etc. And technically, there can only be one Caliph at a time since he speaks for all Muslims, but there have been times when multiple people claimed the title (just like with the Papacy).