Isma'ilis are a Shi'a sect and therefore firmly uphold the Shi'a doctrine of the imamate (imamah). Like their Twelver (Ithna-Ashari) Shi'a brethren, the Isma'ilis regard Ali bin Abu Talib as spiritual successor to Muhammad. After Ali, a number of imams are recognised through to Ja'far as-Sadiq. After Ja'far, the Shi'a divided over who which son should succeed him with the Twelvers following Musa al-Kazim and others following the eldest son Isma'il, hence the term Isma'ili.
Isma'il's life and the events leading up to Ja'far's death are surrounded in mystery and controversy. There are differing accounts of whether Isma'il had the right to succeed his father [1]. He was succeeded by his descendants Muhammad, Abdullah and Ahmad, during a period in which they undertook secret propagation of the Isma'ili faith from their base in Salmiyah in present-day Syria. The propagation was carried through Islamic lands by missionaries (da'i). It is during this time that the Isma'ilis believe that the monumental encyclopedia, the Epistles of the Brethren of Purity (Rasa'il al-Ikhwan as-Safa), were written under the guidance of their imams.
The 11th Isma'ili imam Abdullah al-Mahdi established the Fatimid state in North Africa in the early 10th century AD which by the time of the 14th imam al-Mu'izz had spread to Egypt. In Egypt, the city of Cairo (al-Qahira lit. "the victorious") and the al-Azhar university-mosque were established on the outskirts of al-Fustat. During the time of the 14th imam al-Hakim, a group of followers established the Druze sect which survives today in Lebanon and nearby regions. A dispute broke out over the succession of the 17th imam al-Mustansir between his two sons al-Musta'li and Nizar. Al-Musta'li managed to retain political authority in Cairo with the help of the vizier Badr al-Jamali, while Nizar's supporters later established a rival Isma'ili imamate in Alamut, Iran. Al-Musta'li was succeeded by al-Amir whose assassination marked the beginning of the collapse of the Fatimid reign in Egypt.
Today, the Bohras follow the Musta'lid branch of Isma'ilism believing that al-Amir left a young son named al-Tayyib and that the imamate continues in his progeny in seclusion. They also believe that under al-Amir's direction, his representative (hujjah) in Yemen, al-Sayyida al-Hurrah, established the office of the absolute summoner (da'i al-mutlaq) which has managed the affairs of the community from Yemen and later from India [2]. The Nizari Isma'ilis, also known as Imami Isma'ilis, on the other hand, accept Aga Khan IV as their 49th imam.
Isma'ilis affirm the fundamental Islamic belief in God and adhering to the Qur'an and the precepts of prophet Muhammad. As mentioned earlier, they also uphold the Shi'a belief of imamate. In addition, they state that every part of scripture or ritual in Islam has an outer aspect (zahir) and an esoteric aspect (batin).
Notes
[1] The Twelvers generally maintain that Musa who was explicitly chosen by Ja'far as his successor. Some section of the Twelvers however add that Isma'il was initially appointed as the heir but since he had pre-deceased his father, the imamate was then passed on to Musa. A section of the Isma'ilis challenge this position and state that the imamate cannot be revoked and should always pass on from father to son, and therefore Isma'il would be succeeded by his son Muhammad. Another section of Isma'ilis however hold that Isma'il did not pre-decease Ja'far and that Isma'il's apparent funeral during Ja'far's lifetime was an act to protect him from the Abbasids or that it was actually Abdullah's (another of Ja'far's sons) funeral. Yet another group of Isma'ilis later maintained that Isma'il had not died but had gone into seclusion and would return in the future to restore justice on Earth. This last group were known as the Seveners (Sabiyah) but they are not thought to have survived.
[2] The Bohras themselves have split several times with each sect recognising a different da'i al-mutlaq. The largest sect are known as the Dawoodi Bohras.