Finally a question I can answer!

The basis of the Sunni - Shi'ite split goes back to the succession of the Caliph after the death of Muhammed. The Shi'ites were supported of 'Ali to become Caliph, but instead the first Caliph was decided to be Abu Bakr, a long time friend of the prophet.

However, Shi'ites still accepted Abu Bakr, and the rest of the first 4 Caliphs up until 'Ali became caliph in 656. (Correction here: Accepted as in did not engage in active violent rebellion against them. Shi'ites do not believe that the first 3 Rashidun Caliphs are legitimate.) This is where the split happens though.

The prior caliph, 'Uthman, was assassinated in 656. Mu'awiya, the Governor of Syria, was related to 'Uthman and wanted 'Ali's help to find the assassins. Instead, 'Ali refused and asked Mu'awiya to give up control of Syria. This is the start of the first civil war, called the First Fitna.

'Ali lost the Battle of Siffin in 657and agreed to arbitrate the dispute with Mu'awiya instead of fighting it out. This angered some supporters of 'Ali and he was assassinated in 660. Mu'awiya then convinced the firstborn of 'Ali, Hassan, to give up claim to the Caliphate. And so ended the first Fitna.

Mu'awiya ruled from 661 to 680 when he died and was succeeded by his son Yazid. Hassan, true to his promise, didn't contest for the Caliphate. However, Hussein, the younger son of 'Ali did. This was the start of the Second Fitna. (Correction by /u/tuniki : Hassan didn't contest because he had been assassinated in 670. Shi'ites also believe that Mu'awiya broke his promise with Hassan to not name a successor, and so Hussein was justified in attempting to contest the Caliphate.)

As other users stated, Hussein lost the Battle of Kerbala and was martyred, along with his 6 year old son. This is the reason for the current Shi'ite practice of self-flagellation practiced to this day on the holiday of Ashura.

After Hussein's loss, the Umayyad caliphate was assured of its rule until 750. Shi'ites still believe that 'Ali was the proper caliph, and so refused to recognize the Umayyad caliphate.

And that's the historical case to the split! There are some more details I will add later but I'm on my phone right now. Also, sorry for any grammar mistakes, I'll fix them when I get back to my computer.

EDIT: I got to a computer now and can post some links and sources.

I pulled most of my information from "A History of Islamic Societies" by Ira Lapidus.

There is also this interesting report from the Congressional Research Service on the split between Sunnis and Shi'ites. It focuses a bit more on sectarian differences and resulting violence, but very well gives historical reasonings. It can be found here.

I also didn't mention one important difference between Sunnis and Shi'ites, namely that Shi'ites believe that the Caliph should be in blood relation to Muhammed, whereas Sunnis do not. This is another reason they supported 'Ali (The Nephew of Muhammed) over either Abu Bakr or Mu'awiyah.

EDIT II: Spelling

As a follow up question, to what extent is the split perpetuated by Arabic/Persian ethnic conflicts? It seems (although I don't know much about the subject) that Shia's are mainly in areas which used to be strongholds of the Persian Empire. Also, is Shia Islam in any way influenced by Zoroastrianism?

I understand how the Sunni/Shia split happened on the basis of "who would become the next ruler" but I'm confused on how religiously the split happened.

Does anyone have any info? Were there different preachers?

I've heard that the split only became as violent as it is during the period of the Ottoman-Safavid Wars. Is there any truth to this?

The shiites have never had a caliphate they have always been a minority creating trouble