Glad you could make it. It gets complicated as time goes on, but the difference between Shi'a and Sunni began nearly from the start of Islam. Bear with me on this one:

When the Prophet Muhammad died, he left no heir nor any method for how a new leader should be chosen, that leader - the Caliph Rasul Allah or just Caliph - was supposed to act as a religious and political leader. Shortly after the Prophet's death, the tribes which had converted to Islam - and remained loyal to it (more on that later) - elected representatives who would vote for the next Caliph. This body, the shura, basically held a kind of "National Convention" and the first Caliph, Abu Bakr, was chosen through a democratic process.

Not all followers of Islam felt that such a leader should be elected, but rather that Allah had blessed the blood of the Prophet and so his closest male relative should lead the umma (community of believers). That male relative was his cousin and son-in-law, Ali. The group which dissented was called Shi'at Ali or "the party of Ali. The division between Sunni and Shi'a was largely political at this point and wasn't really that destructive. Yet.

Abu Bakr rallied the umma to attack and defeat the tribes which had abandoned Islam (apostasy is a death sentence in most religions anyway), which was a successful way to temporarily pave over the divisions withing the umma. The problem really became exacerbated when Ali was finally chosen as the 4th Caliph (and the last of the Rashidun or 'rightly-guided Caliphs'). Ali's rule was controversial and plagued by political in-fighting and palace intrigue which makes Game of Thrones look positively optimistic. Eventually, Ali was assassinated while praying by a former supporter (who belonged to a different, short-lived sectarian division called the Kharijites).

After Ali's death, there was a civil war within Islam - though it had never really been peaceful to begin with - where the Shi'a followed Ali's son, Hussein, and the majority (or Sunni) followed a man named Muawiyah. It was a lop-sided civil war because Muawiyah bribed, cajoled, or extorted the generals of Hussein's army into switching sides. The climactic battle between Sunni and Shi'a was fought at Karbalah, in m/d/ Iraq. This massacre is commemorated every year in Shi'a lands as Ashura which /u/Box-ception misunderstands; the Shi'a publicly lament the loss and the men self-flagellate opening up wounds on their backs and chest as a show of piety.

As for the differences in belief, there are a LOT of them. Though both technically adhere to the Qu'ran and other attendant support literature of Islams (the surra and hadith for example) they put different emphasis on different laws and rules. A really notable example is that Sunni have a pretty much hard-and-fast rule on not depicting humans or animals in artwork while Shi'a are a bit more relaxed in that sense, allowing some representations, as long as it isn't the Prophet or Ali or any other revered figure.

There's a lot more to it than that, of course. Throughout the history of the Middle East, Sunni Caliphs (and later Sultans when the Turks take over) engaged in periodic repression and purges of Shi'a. For their part, Shi'a Caliphs (and Shahs when some Turks adopted Persian words and customs) also engaged in brutal repression and purges of Sunni. Perhaps the most famous Shi'a caliphate you've never heard of is the Fatimids of Egypt who allied themselves with a fanatical sect of Shi'a zealots who history refers to as the Assassins. These Assassins would target powerful Sunni individuals, Western Christian crusaders, and those Shi'a leaders they felt were unfaithful to a very dogmatic interpretation of their version of Islam. The Fatimids and Assassins eventually fell out with one another, but the Assassins lived on for a few centuries after the end of the Fatimids in Egypt.

The upshot of this whole story is that the two sects have been oppressing and killing each other off and on for the better part of Islam's history, though in some lands Shi'a and Sunni do get along, usually the dominant sect goes out of its way to repress and purge the non-dominant one even today - just look at the relationship between Saudi Arabia and Iran, for example.

I hope that helps and welcome to AskHistorians.

/u/vetmicheal has summed up most of it, but he has missed out the most important part, the main divide came with the assassination of the 3rd Caliph, Uthman. His assassination was a result of a seige if Madinah by a faction of muslims who were demanding his resignation over (mostly) administrative issues. After his assassination, Ali was made caliph and 2 groups emerged, one wanted the assassins of Uthman punished and the other asked Ali to settle down the whole divide and let go the rebels. Muawiya was then governor of syria and egypt, he pressured Ali to punish the rebels, and refused to accept Ali as caliph until he dealt with them. 2 years down the line another faction, the kharijities were formed. They refused to obey both Ali and Muawiya and decided to assassinate both of them. Muawiya got sick on the day assassins were sent and didn't go to the mosque but Ali went and was stabbed, and died due to it. Hassan, the elder son emerged as the popular choice, but he negotiated with Muawiya, and accepted him as caliph on one condition; that Muawiya will choose his son or relative as his heir. 6 months later, Hassan was poisoned, after which Muawiya decided to make his son Yazid as the heir. Now most of the muslims in Madinah revolted against this including Hussain, the younger son of Ali. At this point, Hussain received a letter from his supporters in Kufa to to come and settle in Kufa to start a rebellion against Yazid. Hussain began his journey to Kufa but was intercepted by the army of Yazid in Karbala (modern day Iraq) it is here that the family of Hussain was besieged and killed. People of Kufa didn't come to the rescue and after his death came out in numbers and grieved with supporters of Hussain. After that, Shias trace down their Imams, which were their leaders and descendants of Ali. Over the years the Shias developed a seperate jurisprudence. The most prominent different on surface is however is that Shias grieve the death of Hussain each year by beating themselves and other rituals. Sunnis however believe that after the four caliphs, there was no unanimous leader of the muslims and derive their jurisprudence mainly from the works of Abu Hanifa and his contemporaries.

I posted this elsewhere, but it should meet the standards of this community and can give you a quick summary of the underlying cause for the split, as well as a brief overview of the very early but extremely important days of Islamic history that would set the trajectory of this conflict.

The split between Shia and Sunni happened the moment Muhammad died. It had to do with his succession, and who would be the next Caliph. Sunnis believed, that as the Sunnah dictates, the succession process is to be a democratic one. So the Shahaba of Muhammad (think something like an electoral college in this context) nominated Abu Bakr, Muhammad's longtime advisor and uncle-in-law. The Shi'a believed that the caliph is to be divinely appointed, and that blood was the main factor of succession, and so they supported Ali, Muhammad's cousin. Ali himself acknowledged Abu Bakr as Caliph. His supporters did not. The supporters of Ali became the Shia and from there on out the rift existed and has over time widened. But keep in mind, for most of history, the two have peacefully coexisted. The times conflict has erupted, it was over political reasons. Without a fail, every single time. And those political reasons are 'secular' earthly reasons that are not so different from, for example, the Catholic/Protestant clashes.

In Sunni Islam, the Caliph is democratically elected. In Shia, he is divinely appointed.

What I want you you understand is, this is a political-not theological divide. Both Muslims abide by the 5 pillars. Both believe there is no God but God and Muhammad was his messenger. Both read the same Quran. They do have some ritual differences, but these rose from the political divide and not the other way around as we see in the Catholic/Protestant divide.

Abu Bakr was the 1st Khalifah of the Rashidun (Rightly Guided) Caliphate from 632 to 634. All 4 of the Rightly Guided Caliphs were the Shahaba (companions) of Muhammad, and the most revered figures in Islam sans the Prophets and arguably Maryam. The bulk of the Hadith comes from the accounts of these men. Traditional Arab society was very structured around oral tradition, and so things like poetry and quotation were integral parts of their culture. This is the chief reason that the Qur'an was able to survive in its original format all these years- it is first a oral/auditory tome, and a literary one second, though both medium are essential. So we have very clear accounts of stories from the life of Muhammad that shed light on his motives and personality, which is what makes the Hadith significant at all. And it is in large part thanks to these men that these stories have survived so long, and moreover, arguably that Islam ever became a thing outside Arabia in the first place.

What's interesting is, do you know who the 4th Rashidun Caliph was? Ali. But his wasn't a natural ascension. His predecessor, Uthman, was assassinated. By who? No one really knows. One of history's great mysteries. Lets call them 'Rebels'. Rebels surrounded his house roughy 12 years into his rule, and demanded that he abdicate. And he would have done it too, many of his Shahaba advised him to do so. But he couldn't.

"Perhaps God will clothe you with a shirt, Uthman" the Prophet had told him once, "and if the people want you to take it off, do not take it off for them."

So he honored the pledge and was executed. An election was held, Ali's time had come. But he accepted the title very reluctantly, as he rightfully perceived that the events surrounding his ascension was a foreboding symptom of things to come. The first thing he did when he took office was dismiss all the governors from the various provinces throughout the Caliphate. Now, yes, this was not long after Muhammad's death. Ali was his cousin and grew up around the man, so not even one generation had past. Yet, this is what the Caliphate looked like in his time. All the governors accepted, except for one. Muwaiya, the governor of Syria. He refused to leave his post until the assassins of Uthman were brought to justice.

In a separate place, Ayesha, the wife of the Prophet, was on a pilgrimage to Mecca when she heard about Uthmans death. She was so infuriated, she turned right around, with an army assembled, and set out to demand Ali do Uthman justice. But Ali was in a really tough predicament: he didn't know who was responsible. We still don't know who was responsible. So he refused to decree a punishment without proper evidence. Ayesha and Muwaiya found this unacceptable, and so their supporters and Ali's Army fought. This was the first time in history that Muslims took up arms against Muslims, in The Battle of the Camel.

Ali decisively won. When he and Ayesha next met, they reconciled and Ayesha saw the error in her judgement. She was sent back to Madinah under military escort. She never forgave herself.

You'd think the story has hit a climax, but it still hasn't. Towards the end of Ali's rule, the Kharijites emerged. These were a group of fanatics who rebelled against the Caliphate and claimed that all rulers were illegitimate, that the only ruler mankind needs is God. They are hailed as, and in fact were foretold by Muhammad as, one of the most dangerous enemies of the Ummah. The Khaijites dispatched 3 sets of assassins to kill Muwaiya, Amar Bin Al-Aas (governor of Egypt), and Ali. The first two failed. The last found Ali in the Grand Mosque of Kufa on the 19th Day of Ramadan. Ali was slain by a poison-coated sword while in prostration. Thus, the Rashidun Caliphate came to a bloody end, at the hands of terrorists.

Why did it come to an end? Because, Muwaiya, who succeeded Ali, APPOINTED HIS SON, Yazid, as his successor. And Yazid successfully secured his rule by having all other challnegers either subjugated, or as in the case of The Prophets own Grandson, murdered. This, in effect, gave birth to what the world would forever know as, the Ummayyad Caliphate. But see, remember what I said about the Caliph is supposed to be democratically elected? The Umayyad and Abbasid Caliphates were both hereditary dynasties. One of the greatest irony's and probably a TIL for most who have made it this far:

TL;DR: The Umayyad and Abbasid Caliphates were not Islamic states, nor were they even rightfully Caliphates.

As a followup question, how many people were Muslims at the time of the death of the prophet Mohammed?

I mean, to have this much division at the inception of the religions makes it seem ridiculous for it to have lasted out of the incubator.

Also can someone explain Ibadi to me?

What about nation of islam?