The Shi’a call themselves so, because we describe ourselves as “Shi’an-e-Ali” (Party/Followers of Ali – “Shia” means party member or follower). Now this difference of belief did not immediately result in a division amongst Muslims at that time. The division occurred later when the self-appointed caliphs of the Islamic nation began persecution of Shi’as because of their refusal to acknowledge the caliphate of these rulers.
These Shi’a refused to acknowledge the caliphate of these rulers because of the Shi’a belief that only the Prophet (under instructions from Allah) could appoint a caliph for the Muslim ummah, as he had already done so during his lifetime.
The history of the Muslim ummah, after the demise of the Prophet, unfolded in the following manner. After a large portion of Muslims refrained from acknowledging Imam Ali as the first caliph, he (Imam Ali) chose to wait with patience over this affair, rather than fight for his right, since the Prophet had advised him in the last days of his life to deal with the clamities that followed his (the Prophet’s) death with patience, so as to not cause divisions within the Muslim ummah. As a result, he withrew himself from active participation in political affairs and with matters regarding the handling of the government.
After the death of the third caliph, the state of the Muslim nation had severely deteriorated since the time of the Prophet, and much of the injustice and suffering that was prevalent during the pre-Islamic era had creeped back into society. Muslims, acknowledging the dire situation their nation was in, then earnestly appealed to Imam Ali to take up the caliphate as they realized that there was no one more worthy than him for that position. After a lot of persuasion, Imam Ali reluctantly agreed.
Now within the rule of Imam Ali, Muawiyah bin Abu Sufyan rose up and waged a bloody war against him. He (Muawiyah) unjustly claimed that Imam Ali was either responsible or connected to the death of the previous caliph, and that this war of his was revenge for that death. Eventually, after a long war, a compromise was reached, and partial peace was restored.
However, after the death of Imam Ali, Muawiyah declared himself caliph of the Muslim ummah, and confronted Hasan bin Ali (a.s.), the son of Imam Ali, who was appointed by him as his successor. Again Muawiyah tried to wage a war against the true caliph, but this time Hasan bin Ali (a.s.) managed to work out a compromise with him that permitted him governance of the Muslim ummah till his death, after which it would return to Hasan bin Ali (a.s.) or his successor.
However, before Muawiyah died, he appointed his son Yazeed bin Muawiyah as the next caliph, blatantly contradicting the terms of the peace-treaty that he had given his oath of allegiance to earlier with Hasan bin Ali (a.s.). Meanwhile Hasan bin Ali (a.s.) had appointed his brother Husayn (a.s.) as his successor, and he was thus the rightful caliph. But Yazeed had taken that title for himself, and ordered Husayn (a.s.) to pay allegiance to him. Husayn (a.s.) refused. As a result, Yazeed’s forces salughtered Husayn (a.s.), the grandson of the Prophet (pbuh&hp), and his family members in the land of Karbala, on the day of A’shurah.
In Yazeed’s rule, and in the rule of the succeeding rulers appointed by him, the Shi’as suffered extreme levels of injustice and mistreatment from these rulers, who deliberately persecuted Shi’as without inhibition, and without any regard for Islam or morality. As a result the Shi’as withdrew themselves from the government (which was mainly Sunni), and remanied secluded from matters of caliphate for hundreds of years that followed, and this seclusion has resulted in the prominent Shi’a-Sunni split that you see today.
For a much more detailed analysis of the historical causes of this issue, read the following online book: