To understand this, one must first understand the concepts of Imamate and Caliphate in Shi'ite Muslim thought.The learning and wisdom of Prophets belongs to the divine plan of guidance. Its function is to ensure the survival, continuity, purity and vitality of the Deen.
Now if worldly rule is available, it rightly belongs to the Imam as his Divine prerogative. But if the Imamate is denied that right of worldly rule, nevertheless it remains the Imamate, just as a Prophet is a Prophet, whether his people deny him the right of becoming the worldly ruler, or whether they throw him into prison. In the same way, even if the people of the Imam throw him into prison, he is their Master, the Leader of that community.
Imamat and Khilafat, the two offices in Islam, are different in their entirety.
People owe allegiance to an Imam for it is his Divine right. In the case of a Caliphate, people pledge their allegiance and by virtue of it, he becomes worldly ruler or Khalifa. If there is no territory, there is no Khalifa. But an Imam is an Imam - be there territory or no territory. Khilafat or Hukumat can be given or taken away. Imamate, on the other hand, can neither be given nor be taken away. Imamate is above any dispute, or above any controversy. It cannot be disputed as it is so ordained by Allah Almighty. Khilafat or Hukumat can be disputed, and is often disputed. Like the prophets, Imams are also infallible.
Unlike the Sunnis, the Shi'a believe that after the death of the Holy Prophet (PBUH), it was the exclusive prerogative and right of Hazrat Ali (AS) and his male progeny from Hazrat Bibi Fatima (AS) to become caliphs of the Islamic world, because Imamate had already been divinely bestowed upon them.
The Split: how the Ismailis evolved
The Ismailis, like the the Twelver Shia, accept Hazrat Ali, Hazrat Hussein, Zayn-al-Abidin, Muhammad al-Baqir and Jafar Al-Sadiq as the legitimate Imams. However, they differ on the Imamate of Musa al-Kazim, as in his place, they consider Ismael, the elder son of Imam Jafar, who died in his life time in 755, as the actual Imam. And after him, they accept the Imamate of his son Muhammad, who died during the reign of Abbasid Caliph Harun Al-Rashid.
From then on, the Ismaili Imamate continued as such. In the meanwhile, the Ismailis succeeded in establishing the Fatimid Caliphate in Egypt in 934. Now the spiritual head of the Ismailis, besides being their Imam, also took reins of the state as Caliph.
The Second Split
It took place in 1094 AD after the death of of the sixth Fatimid Caliph, Mustansir Billah (who was the 18th Imam as well) when the Regent Al-Fazl placed his younger son Al-Mustali on the Egyptian throne. This was contested by the elder son Nizar, who was defeated and died in prison. Thus, the Ismailis were divided in two severely opposed groups: the Nizaris and Mustalis.
The Third Split
The Nizari Ismailis were further divided in two parts over the succession of Imam Shams-al-Din Muhammad (the 28th Imam). One group followed his younger son Qasim Shah as Imam while the other followed the elder son Muhammad Shah. These groups, thus, became known as Qasim Shahi Nizari Ismailis and Muhammad Shahi Nizari Ismailis. The family of Agha Khan belongs to the Qasim Shahi Nizari Ismaili branch, and the head of this line is called the Qasim Shahi Nizari Ismaili Imam.
In the dispute over the succession of Caliph Mustansir, Hasan-i-Sabah, who was operating as an Ismaili da'i in Persia, upheld the cause of Nizar and severed his relations with the Fatimid regime. He founded his own independent Nizari Ismaili state in 1090, which was headquartered at the fort of Alamut. Hasan-i-Sabah, known as the Old Man of the Mountain, and his followers as Assassins, was hated by the Twelver Shi'ites, the Sunnis, the Seljuk Turks, the Fatimid Egyptians and also by the Abbasid Caliphate at Baghdad.
His Fidayeen wreaked havoc on the Muslim world, assassinating prominent personalities such as Nizam-al-Mulk, the Grand Vizier, Al-Borsoki, the powerful king of Mosul and Aleppo and other influential clerics who were preaching jihad against the Crusaders from Europe, who were expanding fast on the eastern rim of the Mediterranean Sea. The kingdom established by Hasan-i-Sabah was ruled by three da'is and five Imams for 166 years till its destruction at the hands of Hulegu Khan in 1256.
The family of the Qasim Shahi Nizari Ismaili Imams settled in the Mohallat area of Persia. Its 46th Imam, Hasan Ali Shah, was appointed as the governor of Qum by the Qajar ruler of Persia, who bestowed on him the honorific title (laqab) of Aga Khan (less commonly but more correctly translated as Aqa Khan), meaning Lord and Master. Henceforth Hasan Ali Shah became generally known as Aga Khan Mohallati, because of his royal title and the family's deep roots in the Mohallat area. The title of Aga Khan remained hereditary among his successors, the Qasim Shahi Nizari and imams of modern times.
After Qum, Aga Khan I was appointed the governor of Kerman, where because of an unmanageable rift with the Qajar Emperor, he had to flee to India in 1841. This marked the end of the Persian period of the Qasim Shahi Nizari Ismaili Imamate, which had lasted seven centuries since the Alamut era. Aga Khan I, the 46th Imam, died in 1881 after an eventful Imamate of 64 years. His successor, Aga Khan II died in 1885, after an Imamate of only four years. Then came the third Aga Khan, Sultan Muhammad Shah, who gave the real boost to the image, stature, prestige and well-being of the Ismaili community during 72 years of his Imamate. He frequently visited his community at places like Zanzibar, Mombasa, Dar es Salam , Nairobi, Kampala and Tanga and always maintained excellent relations with the rulers and elites of Great Britain, France, Turkey and Russia.
The place and stature of the third Agha Khan
He was appointed Grand Knight Commander of the Indian Empire and was given a seat the Viceroy's Council.
He headed the Muslim delegation that met Lord Minto at Simla, asking the Viceroy to regard the Indian Muslims not as a minority but as a nation within a nation whose members deserved adequate representation on both local and legislative councils of the land.
He joined in the founding of the All India Muslim League in 1907, and served as the permanent President of that body until he himself resigned from this position in 1912.
He played a leading role in the elevation of the Muhammadan Anglo-Oriental College at Aligarh to university status. He was accorded in 1916 the status of a first-class ruling Prince of the Bombay Presidency, although, unlike other native rulers of India, he did not possess a territorial principality.
In 1928 he presided over the All-India Muslim Conference held at Delhi, which was to formulate views on how independence should evolve for India. In 1937, he was elected as President of the League of Nations for a session.
Like his grandfather, His Highness Price Karim Aga Khan IV (known to his Murids as Mawlana Hazar Imam) is administering the affairs of his community in a commendable manner since 1957. The Ismailis are invariably well educated, well looked after, prosperous and peace-loving people who are spread all over the world.
Before concluding, let me briefly state main tenets of the Ismaili faith. Like all Muslims, Ismailis affirm the unity of Allah Almighty, followed by that of Divine guidance through the messengers, of whom the Holy Prophet (PBUH) was the last. Thereafter, the Ismailis comprehend Islam through the guidance of the Imam of the time, who is the inheritor of the Prophet's authority, and the trustee of his legacy.
The Imam has the authority to make the outward form of the religion relevant to the needs of the time and ensure that his followers strike a balance between spirit and intellect and explore the frontiers of knowledge through scientific and other devices as was done in the past by the prominent Ismailis such as Ibn e Hasham, Ibn e Sina and Al Tusi. In Ismaili thought, the Imam is the only one who is bestowed with the power and wisdom to understand the inner meanings of the Ayats of the Holy Quran while the general public knows and follows the outer meanings (zahir) only.