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A short (believe me, I tried to be as concise as possible) answer to a very broad question is:

After the Prophet Muhammad died (the traditional date, sometimes questioned by modern historians) in 632 C.E., the succession of leadership went to Abu Bakr, a well known and respected companion of his. This succession didn't go uncontested, as a number of people thought that Muhammad had already explicitly named a successor through his son-in-law and first cousin Ali ibn Abi Talib, but it seems that Ali himself did not immediately press the issue, since Abu Bakr also had the support of Umar al-Khattab, another famous companion of Muhammad's. Abu Bakr titled himself as Khalifat Rasulillah, or the "successor/deputy to the Messenger of God" and was the first Rashidun Caliph. After Abu Bakr came Umar, then Uthman (who was from the Umayyad family, but had taken Muhammad's side during the early years of Islam) and the finally Ali. These four were all Arabs from Mecca, all belonging to the tribe of Quraysh. Ali was Muhammad's own blood, belonging to the Hashimid family within the tribe of Quraysh. The Umayyad family was a distant cousin family to them.

During Ali's caliphate there were several issues, the biggest of which was an Umayyad rebellion due to unhappiness with how Uthman's caliphate had ended (Ali never punished Uthman's assassins, which to the Umayyads was as good as official sanction, though traditions indicate he had sent his son Hussayn to protect Uthman during his last days). The Umayyad rebellion was led by Mu'awiya, the governor of Syria.

In any case, Ali was killed while praying in Kufa, and Mu'awiya quickly seized the reigns of the Caliphate. This was the beginning of the Umayyads, who then ruled from Ali's death in 661 C.E. to 750. The Umayyads were Arabs, and very pro-Arab in their policies. They adopted the ruling customs of many of the structures left behind by previous conquests, and it took a while for them to introduce Arabic as the government language. They did not encourage conversion, and sought to keep the ruling Islamic clique as Arab as possible. This led to a lot of issues with regions in Mesopotamia and Persia, which had many Persians (and one generation later, mixed ethnic Persian/Arabs or Persianized Arabs) and was a hotbed for anti-Umayyad sentiment. Particulaly strong also was Shi'a discontentment, since the Umayyads were not very kind to Ali whom the Shi'a saw as the rightful Caliph. The Shi'a supported various sons of Ali in the wake of Ali's death - some supported Hasan ibn Ali, who did not take up arms against the Umayyads. When Hasan passed away, they supported Hussayn, who DID take up arms, and was massacred with his entire family save for one son and one sister. These two brothers were special in that their mother was Fatima, Muhammad's daughter, unlike some of Ali's other sons through other wives. Yet others still rallied around a son like that, a nephew of Muhammad rather than a grandson through Fatima, named ibn Hanafiyya, this rebellion too was defeated. Shi'a sentiment remained powerful in Mesopotamia, particularly in Kufa.

The Abbasids relied on the pro-Arab and anti-Shi'a policies of the Umayyads to gather support among Persians, non-Muslims, and Shi'as to launch their rebellion in Khorasan. The rebellion was very secretive, and only when it was sprung was the identity of their candidate, the future Caliph as-Saffah, revealed. The Abbasids rose in revolt in 747 and by 750 had destroyed the Umayyads almost completely. However, one Umayyad managed to escape to Spain and set up an Umayyad caliphate there.

The Abbasids fixed many of the issues that plagued the people who had risen in their favor, but not the issues of the Shi'a. As-Saffah was similarly related to Muhammad as the Alids were, as As-Saffah's ancestor Abbas was Muhammad's uncle, like Ali's father. However, Ali's sons were Fatima's sons as well, and thus directly descended from Muhammad, rather than from an uncle or so like the Abbasids. Shi'as were still angry and were upset that their head, Ja'far as-Sadiq, had not become Caliph.

Ja'far as-Sadiq lived a peaceful existence as the head of the majority of Shi'as, He was the great-grandson of Hussayn ibn Ali, through Hussayn's son Zayn al-Abideen who survived the Karbala massacre. Ja'far died in 765, but his death was followed by another succession issue. His designated heir, Isma'il, had either disappeared or died before his father, and there is no concrete evidence that Ja'far ever nominated another successor. This is a thorny issue, since the Shi'a concept of Caliph and Imam is ma'sum, or infallible, meaning the Caliph/Imam can do no wrong. How could Ja'far appoint a son who died before him? The only answer to some was that Isma'il had become Imam, and that his son Muhammad was now the next Imam. Others believed that Musa Kadhim, another son of Ja'far's, had become Imam. Muhammad ibn Isma'il's supporters became the Ismaili Shi'a, and Musa's supporters were the Ithna Ashariyya, or the Twelver Shi'a.

Ultimately, Musa's line did not achieve a caliphate, since the Twelver Imams were more or less political prisoners within the Abbasid state. Shi'a sources say that they were poisoned by the Abbasids. The last Imam went into hiding, and Twelver belief to this day is that the Twelfth Imam will return as al-Mahdi, or the savior, to redeem his people.

The Ismailis on the other hand took a page out of the Abbasid revolutionary book, went underground, and began a caliphate-wide conversion mission. At the break of the 900s, the head missionary of the group, a fellow named Abdullah (almost certainly NOT his real name, since the Ismailis used non-descript revolutionary names like the Abbasids had) broke cover and said that he was al-Mahdi and the great-great grandson of Muhammad ibn Isma'il. By this time the Abbasids had lost some power and were less centralized, and the rulers in Africa in their name were unable to fight back against the army the Ismailis had amassed by converting the Kutama Berbers to their cause. For the only recorded time in history, there was a powerful Shi'a caliphate in existence as the Fatimid Caliphate. Al-Mahdi's descendants ruled well into the 1100s but power really left the hands of the Caliph-Imam after the death of al-Mustansir. al-Mustansir's son Nizar was his designated successor, but palace intrigues led to the vizier supporting Nizar's younger brother Mustaali as the Imam. The vizier (name escapes me, I believe it was Malik al-Afdal) imprisoned Nizar and had him executed, but supporters of Nizar claimed to have smuggled Nizar's son to Persia. A string of powerful viziers ruled the caliphate and the Imam was more or less a figurehead. The Nizaris might have gotten their revenge, because al-Afdal was murdered, but Ibn al-Qalansi writes that it wasn't the "Batinis" (another word for the Ismailis) who were responsible. Under that authority they set up the Nizari state in Persia, and launched the Alamut period of Nizari Ismailism. al-Mustali's successors ruled in Egypt until Salah al-Din Yusuf ibn Ayyub put an end to the Fatimids in the 12th century.

The Abbasids by this time had lost a great deal of power (for the same reason as the Fatimids - the loss of centralization, concessions to local rulers and emirs, and the reliance on Turkish mercenaries and soldiers and governors that weren't always the most reliable. However, the Abbasids continued to exist well into the middle 13th century until the Mongols wiped them out with the sack of Baghdad. The Mamluk Sultanate that sprang up in Egypt around that time (1250's and 1260's) kept an Abbasid Caliph in Cairo as a religious figurehead, but the power remained with the Mamluk Sultan.

Beyond the sack of Baghdad my knowledge of Islamic history is limited, so I will defer to others.