The origin of the Sunni-Shiʿa split has both a political and a religious component. It's immediate cause was linked to a succession crisis between ʿAlī ibn Abī Ṭālib and Abū Bakr as-Sadīq.

Following the death of Muhammad, Muslims in Mecca came together to elect a new leader, as Muhammad had not left clear instructions for who should succeed him. The two main candidates were ʿAli ibn Abī Ṭālib, the nephew of Muhammad, and Abū Bakr as-Sadīq, one of his most trusted companions and Muhammad's father-in-law. Abū Bakr, supported by other influential Muslims (such as ʿUmar, the second Caliph) was eventually decided upon to succeed Muhammad. Sunnis here will claim that all the Muslims agreed to support Abū Bakr, but the Shiʿa reject this, and claim that ʿAli was the sole true successor of Muhammad.

The political split then came to a head after those who are known as the Rashidun Caliphs, Abū Bakr, ʿUmar, and ʿUthman. In 656, ʿUthman was assassinated. Muʿawiyya, the Governor of Syria who would later found the Umayyas Dynasty, took advantage of this fact to challenge ʿAli for the Caliphate. After the Battle of Siffin in 657, ʿAli agreed to forfeit his claim and ceded the Caliphate to Muʿawiyya. Later, in 661, he was assassinated at prayers in Kufa.

The next crucial step, and breaking point for Shiʿa, came in 680 when Muʿawiyya died. Muʿawiyya passed on the Caliphate to his son, Yazid I. Husayn, the son of ʿAli, believed that this broke the agreement between his father and Muʿawiyya and raised an army to challenge Yazid. That same year, at the Battle of Karbala, Husayn's forces were defeated and Husayn himself killed. This battle is still marked in Shiʿa communities by the celebration of ʿAshūrāʾ, a commemoration of Husayn's martyrdom. This battle was the last major attempt of relatives of ʿAli to seize the Caliphate. As Ira Lapidus writes, "ʿAli is the ancestor of Shiʿism; Husayn is its martyr" (A History of Islamic Societies, 49).

So in one sense, the origins of the Sunni-Shi'a split came from a political succession crisis. In fact the term Shiʿa comes from the term shiʿat ʿAli, or the Party of ʿAli, meaning they supported his claim for the Caliphate. They disliked the grandiose nature of the court of Muʿwiyya, its increasing rigidity akin to the Persian and Byzantine traditions, and its reliance upon a small section of the Arab tribal elite.

However, we shouldn't consider the dissent over the Caliphate entirely political. One of the other major break-away groups of early Islamic history were the Kharites who declared that there should be no set line of governance and that it should go to the most qualified Muslim. They took as a rallying cry "rule belongs to God alone", which as one can see has clear religious tones. The Kharijites were religious fanatics, believing that they were the only true Muslims. Indeed, it was a Khariji who assassinated ʿAli.

The Shiʿa themselves also had a religious belief central to their political claim. The Shiʿa hold as a core philosophy the necessity of an Imam. In every age there is an Imam, even if we do not know where he is (which would later develop into the idea of an Imam in occultation). This belief is summed up by the Zaydi al-Qasim ibn Ibrahim who wrote:

the desire for sex and food is implanted in men and, if there were not someone to limit and curb it, people would ght against each other to satisfy their desires, and consequently the world would be destroyed ... People need a guide to teach them these restrictions, and this guide is the Imam. Also the Imam punishes people if they disobey him, and rewards them if they obey him. In this manner people are kept safe.

(qtd. in Black, A History of Islamic Political Thought)

It would also be erroneous to believe that Sunni and Shiʿa thought are two distinct and different pillars of thought. They share many of the same principles and there was a large amount of adoption between Shiʿa and Sunni belief. The above justification for the necessity of an Imam was later applied by Sunnis to justify the State, as summed up in the Circle of Justice (in short, the belief that the ruler had the authority to be sovereign. This authority came from his adherence to and application of the Sharīʿa. It was enforced by the King's Army, which was supported by the taxes of the residents of the State. The residents then agreed to pay taxes because the Sharīʿa was enforced, and thus the cycle repeats.).

This is of course only scratching the surface of Shiʿa thought, but it gives you the idea that it is both immediately political, arising around ʿAli himself, and more intrinsically religious, being linked to the conceptions of necessity for a righteous leader for Muslims. If you have any other questions, feel free to ask.

EDIT: Expanded my comment to account for the other comments being deleted.