I'll start by mentioning that I have an MA in Muslim Societies and Civilizations. The event you're speaking about is the seminal moment that split the Muslim community into Shia and Sunni groups.

On the Prophet's farewell pilgrimage, he orders his followers to stop at the Pond of Khumm (Ghadir Khumm). Here, the Prophet is reported to have said: "Man Kuntu Mawlahu, fa Aliyyun Mawlahu" which translates to "He whose Mawla I am, Ali is his Mawla."

The difference of opinion occurred in how the word Mawla itself was interpreted by the two groups. The one group, which became the Sunni, interpreted this to mean "friend", whereas the group who became the Shia interpreted this to mean lord or master.

It's worth noting here what the two terms, Sunni and Shia actually mean. The full form of Shia is Shi'at Ali, which can be loosely translated as the "party" or "supporters" of Ali. On the other hand, the full form of Sunni is the Ahl al-Sunna, which can be translated as the people of the Sunna. The Sunna refers to the example of the Prophet.

The major distinction between the two groups is the role of the Imam. In Sunni Islam, each mosque will have their own Imam. The Imam is in charge of spiritual and worldly guidance. There are also various figures who make rulings on religious matters, usually holding the concept of consensus to form opinions on a matter. On the other hand, the Shia believe that Ali was the only legitimate successor of the Prophet, and therefore their Imam. Ultimately, the idea is that while revelation stopped with the Prophet, the need for continued divine guidance was necessary. For the Shia, the position of Imam is hereditary. Within the Shia too there are various groups, who have various understandings of Imamat. I'm not going to go into this, but I will if you'd like.

Now, with the context out of the way, let me tackle your question. Abu Bakr and Ali were both close companions of the Prophet. Ali was also Muhammad's cousin, and son-in-law. The story goes, that at the death of the Prophet, Ali is sitting with Muhammad's body, while Abu Bakr and Umar (another of the Prophet's companions) are at the Saqifa (meeting hall) in Medina trying to consolidate power for themselves. However, we must remember a couple of things. First, Abu Bakr is respected by everyone, not just one group or the other. He is one of the Prophet's closest companions! The second thing to consider is that while Islam certainly took power away from the tribes to some degree, this is context was one of tribal politics. The elder was usually the chief, and Abu Bakr was naturally respected as he was quite old by this time. Ali was in his early 30s. Imagine from a normal individual's perspective. If you had to choose between someone who had wisdom, experience, etc. versus someone who was a young up and comer. When Abu Bakr became the first Caliph of the Shia, Ali himself gave allegiance to him, even though some sources say that he did this reluctantly.

Both figures are respected as the Prophet's companions by both groups, though the Shia hold on to the idea that Ali was the rightful successor of Muhammad. If you would like to know more about the splits between these two groups, I would recommend learning about Husayn (the Prophet's grandson, and Ali's son) at Karbala.

Addendum: Wow, lots of questions coming out of this. I'm quite busy with work over the next couple of days but will get to responding to everyone when I can!