This is a question that divides Sunnis and Shiites. Here are some answers from contributors:
Answer 1
Politically, it was the First Caliph Abu Bakr.....
Genealogy wise, his son-in-law, Ali bin Abu Talib....
Answer 2: Shiite View
Imam Ali ibn Abi Talib (as) was appointed at Ghadeer Qum according both Shia and Sunni sources at the final major sermon of the Holy Prophet Muhammed (pbuh). However, during the funeral procedures of the Prophet (saaw), whilst Ali ibn Abi Talib (as) was washing the body of the Prophet (saaw) and preparing the funeral, some companions secretly met at the house of Saqeefa. Umar ibn Khattab appointed Abu Bakr and later, as history books tell us, Umar ibn Kattab proclaimed that whoever goes against this appointment, he would 'cut their tongues off'.
Hence, although Ali ibn Abu Talib was the most knowledgable of the companions (there is no doubt about this - we do not see even a fraction of the depth seen in the writings of Ali ibn Abi Talib in any other companions). Hence, the divine knowledge that Muslims believe the Prophet had was taught by the Prophet (saaw) to Imam Ali (as).
Answer 3: Sunni View
The first thing to keep in mind is that the Prophet-hood not Kingship in which the son succeeds the father. The Prophets have no inheritance. The holy Prophet (SAW) used to lead the Prayers till he fell ill and was not able to lead the Muslims in prayer. He deputed this duty to his closest friend, father-in-law, and the first adult Muslim whom Almighty Allah describes as his friend in the Cave of Thor - Hazrat Abu Bakr (RAU). It may have been an indication that he was the most suitable person after him to lead the Muslims. However, he didn't mention any clear name of his Caliph in this regards. It left it to his Companions whom he considered to be equal to the task.
The Ansaar (the Helpers-native people of Madina Munawra) were discussing the matter of caliphate in Saqifa Bani Sa'adia when Hazra Abu Bakr RAU, Hazrat Omar RAU and Hazra Abu Obaida RAU reached there and after a heated discussion Hazrat Abu Bakr (RAU) was chosen the First Rightful Caliph.
On his death bed, Hazrat Abu Bakr (RAU) suggested the name of Hazrat Omar (RAU) to succeed him. The Companions rau agreed to his proposal and Hazrat Omar (RAU) became the second Rightful Caliph.
When Hazrat Omar (RAU) was fatally wounded as a result of a conspiracy, he suggested the names of six holy Companions (RAU) to select a Caliph amongst them. Some of the Companions suggested to him that he should nominate his son Hazrat Abdullah bin Omar as Caliph. He didn't agree to this proposal. The six companions nominated by him were: Hazrat Usman ibni Affan, Hazrat Ali ibni Abu Talib, Hazrat Zubair bin Awam, Hazrat Talhah bin Obaid Ullah, Hazrat Abdul Rahman bin Aof, and Hazrat Sa'ad bin Abi Waqas rau all of them. After some sittings and discussions, Hazrat Usman bin Affan rau was chosen the third Rightful Caliph.
During the last days of Hazrat Usman rau rebels sieged his house and killed him. Then they forced Hazrat Ali rau to become the fourth Rightful Caliph. Yet, his Caliphate was never completely established till he was killed by a Khawaji. After his death His elder son Hazrat Hassan rau became the caliph but be withdrew his name, made peace with Hazrat Muawiya rau.
It is interesting to note that those who claim that Hazrat Ali rau was the real successor of the holy Prophet (SAW) can't give any logic against this fact that during the Caliphate of first three Rightful Caliphs, Hazrat Ali rau had been the honorable member of the Advisory Council (Majlis-i-Shura). He had been an ardent supporter of these Caliphs. When the rebels sieged Hazrat Usman's house, he sent his sons to guard the house of the Caliph. He and his sons had been receiving stipend from these Caliphs. Hazrat Shahr Bano was given in the wedlock of Hazrat Husain rau by Hazrat Omar rau. All these actions become Haraam, then.
Was Hazrat Ali rau such a coward to remain true to them, and not to rise against them? Or was a hypocrite? Was he not as brave as his son Hazrat Hussain rau who rose against Yazid? He was the lion of Allah, one of the most dauntless persons. He is respected by all Muslims of all sects. The love of Ahli Bayat is a part of the Eeman of all Muslims. Yet, history shows that if Hazrat Abu Bakr rau and Hazrat Omar rau had not been selected Caliphs, Islam would not have spread to that extent. Hazrat Ali rau was much younger for this uphill task that needed to keep the Muslims united under one banner. He could not establish even his own Rightful Caliphate completely. All the stories concocted later on have no sound proof. Whatever message our beloved Prophet (SAW) received from Almighty Allah was in toto, completely preached by the Prophet (SAW) He didn't hide anything. Those who accuse that the Prophet (SAW) didn't convey all message should be mindful about their Eeman. In his last sermon at Arafaat he rose his finger thrice and said, "O My Allah, you also bear witness that I conveyed to your people all what you revealed to me."
Hi a simple answer Shiites say: Prophet Muhammad has chosen Ali as his successor, but Sunnis say that has not happened.
It's because Shiites beleive that Muhammad's son in law and follower Ali, shud have been The Prophet's successor rather than his closest companion (Best Friend).
Muhammad was not the founder of any branches of Islam. In fact, he was not even the founder of Islam, rather the messenger. As well as this, Muhammad and Islam discourage branches within Islam so it clashes with the question.
Abou Bakr Assidiq was elected by Muslims as prophet Muhammad (PBUH) successor of the prophet death.
The Shiites believe that only descendants of prophet Muhammad (PBUH) can lead Muslims. refer to related question below.
Caliph
Shiites.
Hazrat Umar (R.A)
Caliph (or Khalifa in Arabic)
What caused the split was Muhammad's death in 632.
Caliph (or Khalifa in Arabic)
i think he should