To keep it simple..

After the death of the Prophet (Peace be upon him) the Muslims agreed to have a Caliphate, and Abu Bakr, who was the closest to the Prophet (pbuh) during his lifetime, was given allegiance as leader by the Muslims at the time. Almost all the Muslims at the time found nothing wrong with this. These Muslims later became classified as Sunnis.

However, a quite insignificant minority at the time was totally against the choosing of Abu Bakr as the leader of the Caliphate. In fact, they thought no one, other than Ali (who was a cousin of the Prophet (pbuh), also a very close companion) should lead the Muslims. This is because they strongly believed that Ali's right to lead was divine. This minority of the time became classified as Shias. They believe that the Muslims that gave their allegiance to Abu Bakr (which was almost all of them) are traitors to Ali, and even to the Prophet (pbuh)

Ali, later, did become the leader of the Muslims. In fact, he was the 4th Caliph of Islam.

It was after the assassination of Ali, that Shias became significantly important. In the years to follow, battles were fought and parallel caliphates were run for both group of Muslims. Shias had leaders amongst them who they claimed were divinely inspired by God. These men are called Imaams. So the basic and most fundamental part of the Shia creed is that only Imaams can rule over the Muslims. Ali was the first Imaam, and God continues to send Imaams till the end of time.

Sunnis are completely against this as they believe that no one after the Prophet Muhammad (pbuh) has divine importance, whether they belong to the family of the Prophet or not.

Because of these clashing beliefs, both Sunnis and Shias have always considered the other party to be in the wrong path and themselves to be "the true muslims". This is part of the reasons why Sunnis and Shias continuously fight against eachother, and sadly, even to this day.