Quraish (Arabic: قريش) is the Meccan tribe that Muhammad belonged to. Ironically, it was his own tribe that was his chief opponent for most of his life.
Quraysh was composed of twelve clans. ( ref)
Quraish branched out into various tribes, the most famous of whom were Jumah, Sahm, ‘Adi, Makhzum, Tayim, Zahra and the three septs of Qusai bin Kilab: ‘Abdud-Dar bin Qusai, Asad bin ‘Abdul ‘Uzza bin Qusai and ‘Abd Manaf bin Qusai.
‘Abd Manaf branched out into four tribes: ‘Abd Shams, Nawfal, Muttalib and Hashim. From the family of Hashim came Muhammad ibn Abdullah ibn Abd al-Muttalib ibn Hashim. ( ref)
A list of them follows:
Arab lineages allegedly originate from three groups:
Perishing Arabs: These are the ancients of whose history little is known. They include ‘Ad, Thamūd, Tasam, Jadis, Imlaq and others.
Pure Arabs: They allegedly originated from the progeny of Ya‘rub bin Yashjub bin Qahtan so were also called Qahtanian Arabs.
Arabized Arabs: They allegedly originated from the progeny of Ishmael and were also called ‘Adnanian Arabs. The Quraish are a branch of the "Arabized Arabs".
The Quraish had become a prominent tribe in Mecca before the birth of Muhammad and essentially ruled the city. Before Muhammad's birth, the tribe had split into different clans, each with different responsibilities. There were some rivalries among the clans, but these became especially pronounced during Muhammad's lifetime. The message spread by Muhammad was threatening to some clan leaders, and they tried to silence him by putting pressure on his uncle, Abu Talib. Many of the clans also began to persecute the followers of Muhammad, and even went so far as to Boycott them. This response led Muhammad to initially send some Muslims to migrate to Ethiopia, and later would lead to his own emigration to Medina.
After Muhammad's reconquest of Mecca, he pardoned all those who had oppressed him before, and peace among the different clans was maintained. However, after his death, clan rivalries reignited, playing central roles in the conflicts over the Caliphate and contributing to the Shi'a-Sunni divide.
Part of Shi'a belief stems from the fact that Ali was a member of Muhammad's clan, the Banu Hashim. Abu Bakr, while a close companion of Muhammad, came from the Banu Taim clan.
The second Caliph, Umar ibn al-Khattab, was from the Banu Adi clan.
The third Caliph, Uthman ibn Affan, was the Umayyad clan.
When Ali was made caliph after the death of Uthman, the Caliphate was in the hands of the Banu Hashim, but he was almost immediately challenged by Muawiyah, who was a member of the Umayyad clan. After Ali's assassination, the Shi'a hoped his son Hasan would become Caliph, but he deferred to Muawiyah, who established the Umayyad line of Caliphs.
After the death of Muawiyah, his son Yazid became Caliph but was almost immediately challenged by Ali's younger son, Hussein, who would not swear allegiance to Yazid for a number of reasons, among which is that the Caliphate was not supposed to be hereditary. Yazid was a corrupt man but had more military strength than Hussein and Hussein was viciously killed at the Battle of Karbela. This event further added fuel to what would lead to a full schism between Shi'a Islam and Sunni Islam.
The fact that Muhammad's descendants through Ali would be continually persecuted by Umayyad Caliphs did not help the matter.
It seems that initially, the difference between Shi'a Islam and Sunni Islam was simply over politics. Only under the later Abbasid Caliphs would actual doctrinal religious differences arise.