The four founders of Hogwarts School of Witchcraft and Wizardry did not just get together because they were the most powerful wizards of their age. Salazar Slytherin and Godric Gryffindor were great friends, at least for a time.
They shared a mutual respect for one another and worked together for the benefit of wizardkind. This is where the idea of founding Hogwarts came to be. Rowena Ravenclaw and Helga Hufflepuff were also great friends, and so the four of them worked to this end.
Godric and Salazar had visions of shaping the future of the Wizarding World and better defining the position of Wizards towards Muggles. This is where their ideologies clashed, as we know well Salazar had radical notions against muggles and half-bloods even.
Why did Godric Gryffindor and Salazar Slytherin disagree?
Godric’s favor for muggle-born students to attend Hogwarts was the strongest of all four founders. Salazar was against it. They did not become enemies over this, but when Salazar realized he cannot convince the other three founders to only accept pure-bloods, he left Hogwarts.
This was rather soon after the School’s founding in 993. The others still chose to honor him by keeping the Slytherin House name, but they didn’t know that Salazar secretly built the Chamber of Secrets. There he hid the basilisk that was trained to attack and kill muggle-born witches and wizards.
Godric Gryffindor’s Hat was turned into the Sorting Hat
While the gang was still together, they did establish some ground rules for the school. One of the first problems was how to sort the students correctly to their respective houses. This is where they came up with the sorting hat. Godric took off his hat and all four founders enchanted it together.
Only Godric didn’t have heirs
Among the Hogwarts founders, only Godric Gryffindor left no heirs. The other three all left offspring to carry their name and continue their family tradition. We know that Salazar Slytherin’s line had particularly influential wizards, including Voldemort with whom the line ended.
Rowena Ravenclaw’s daughter was Helena. Her ghost is the one that tells Harry about the Horcrux diadem in the final book and movie. She stole her mother’s diadem while she was still alive, sometime in the late 10th to early 11th century. Voldemort later tricked the ghost into revealing the location of the diadem to him.
Helga Hufflepuff also has a line of heirs. She was the possessor of Salazar Slytherin’s locket and Helga Hufflepuff’s cup, two items that Voldemort also collected and made into Horcruxes.
Historical theory about Salazar reconciling with the other three is untrue
There is a popular theory in the Wizarding World history that, towards the end of his life, Salazar reconciled with the other three founders of Hogwarts. This is mentioned on the W.O.M.B.A.T. or Wizards’ Ordinary Magic and Basic Aptitude Test.
The Wizarding Examinations Authority administers this test to all witches and wizards who seek to obtain credentials for working after finishing their studies. However, this theory is likely false, because if Salazar would reconcile and return to Hogwarts, he would likely remove the lethal basilisk.