Edit: Thank you for the gold! I'm glad I provided an answer you enjoyed reading! :)

WARNING HUGE TLDR ALERT:

My time to shine!

Your question is a complicated one and deserves a complicated answer. To begin with, let's do a cursory examination of relevant Islamic history as a whole.

Muhammad passed away in 632, leaving no son as an heir to the community. The majority of Muslims maintain that he did not specify a successor, but a vocal group did. They said that Muhammad had called 'Ali ibn Abi Talib, his companion, first cousin, and son-in-law, to a stage in front of a group of pilgrims and proclaimed "He whose Mawla I am, Ali is also his Mawla". People interpreted that in different ways - mawla can mean anything from Lord to Ally to Friend. Those who interpreted it as Lord said that 'Ali was naturally the next leader of the Muslim community. Not only that, they also maintained that 'Ali was different from the other candidates. Unlike the regular men of Muhammad's clan, 'Ali was elevated - Aaron to Muhammad's Moses, if you will. In their view he possessed a spiritual power that, while not on par with prophethood, exceeded that of Abu Bakr or 'Umar or any of the other candidates for the Caliphate.

Well, long story short, three other guys became Caliph of the Muslim community before 'Ali managed to. 'Ali was the first "Shi'a" Caliph, but at this time there was no group you could truly designate as Shi'a. The word itself simply means "Party" - the formal name of the Shi'a is the Shi'at ul 'Ali - the Party of 'Ali. 'Ali's caliphate was contested, bloody, and tragic. 'Ali fought a war with the relatives of the previous Caliph, and though he had them near defeat at one point, 'Ali chose to make peace rather than continue war. A group of his own followers were so enraged by this seeming betrayal from their leader that eventually, in 661 A.D., they assassinated 'Ali while he was praying in Kufa.

This is where the tumultuous history of the Shi'a truly begins. From this point onwards, the Shi'a start diverging from what will eventually become Sunni Islam - first politically, then theologically. For all intents and purposes, the Shi'a were theologically similar to the Sunni until the mid 700s, when their ideas truly began to diverge.

'Ali's son Hasan, through his wife Fatima, daughter of Muhammad, was first considered to be the next challenger for the Caliphate from the line of 'Ali. However, the governor of Syria at the time, Mu'awiya ibn Abu Sufyan (founder of the Umayyad dynasty) took control of the Islamic caliphate and got a settlement offer with Hasan, who by all accounts was more concerned with peace than war, decided not to contest the Umayyad claim.

This changed with Hasan's brother, Husayn. Husayn was by and large the most tragic figure in Shi'a history, and for good reason. When Hasan passed, Husayn became the head of his family and decided to once again use the 'Alid claim. Iraq, particularly the cities of Kufa and Basra, were staunchly pro-'Alid and anti-Umayyad, and invited Husayn to Iraq and offere to support him against the Umayyads. By this time Mu'awiya was dead, and his son Yazid was in power. Mu'awiya angered many Muslims by making the Caliphate's succession dynastic, instead of through nass, formal appointment, or shura, election by council. Yazid was hated as a drinker, profligate, and all-around bad ruler, and though he certainly was no role model in terms of rulership he wasn't quite that terrible. In any case, Yazid heard of Husayn's march to Kufa from Medina and intercepted Husayn's family at a dusty plain called Karbala, in 680 AD. The Kufans and Basrans did not come to Husayn's aid in time and his entire family, with the exception of his sister Zaynab and his infant son Zayn al-Abideen, were massacred.

Cont. in the reply

The Fatimid Caliphate, which at its height controlled North Africa, Sicily, and the Levant, was Shi'a.

As for why there haven't been as many Shi'ite caliphates as Sunni ones, note that Shi'ites believe that only relatives of Muhammad can become caliph. This obviously limits how many Shi'ites can declare themselves caliph. The Fatimids were descended from Muhammad's daughter Fatimah.

In fact, Twelver Shi'ism (the most popular branch of Shi'a Islam today) is even more strict about who can be caliph. Only twelve people, called the Twelve Imams, can claim to be Muhammad's rightful spiritual and political successors.

There's more information about the Isma'ilis at ismailignosis.com