Sunni Point of view:
This was one of the worst tragedies in the History of Islam. It is because of this tragedy the Muslims have been divided until this day. Indeed this was a disaster.
I think no one can explain and comment on this tragedy better than Shiek Ibn Kathir and shiek al-Islam ibn Taiymiyah.
Ibn Taymiyah said:
Yazid was born when Othman (RA) was the caliph at that time. And he didn't see the prophet or meet him. So he is NOT CONSIDERED A COMPANION with the agreement of all scholars. Nor he was known for being a pious and a good person. But he was one of the Muslim youths. AND HE WAS NOT AN INFIDEL OR LIBERTINE . He took the rule from his father and became the caliph after the death of his father.
Now this is me adding some info:
Muwayia who is the father of Yazid is a companions which means no one is allowed to curse him or talk bad about him because attacking a companion means hurting the prophet Mohamed PBUH.
The Messenger of Allah (PBUH) said:
Do not curse my companions, for by the One in Whose hand is my soul, if any one of you were to spend the equivalent of Mount Uhud in gold, he would not reach the status of one of them, nor even come half way."
(Agreed upon).
Continuation of the quote from Shiekh Ibn Tiymiya:
and Yazid took the rule from his father. However he was a Strong, brave and generous man. Hhe didn't use to make sinful actions in front of Muslims or spread indecencies everywhere as some of his rivals claim. and during his rulings some great events happened one of them is the Death of Hussien (RA).
Now, here were I want you to concentrate. People here try to change the history and cast false information about Yazid.
Shiekh Ibn Tiymiyah says:
Yazid didn't order His soldiers to kill Hussien, he wasn't happy when he heard that Hussien died, he didn’t touch his lips with a rod, nor he commanded to take the head of Hussien to the Palace. However, He ordered his soldiers to stop him from going to Kuffa and make him go back to Madinah. And if he had ordered to kill him then all muslims would have criticized him on this issue.....Then Hussien asked the soldiers either go to Yazid, go to kuffar, or go back to makkah. But they didn't allow him to do that unless they take him as a prisoner. then Omar BIn Sa'ad ordered his soldiers to attack Hussien. so they killed him unjustly along with some of his family and his men. May allah have be satisfied on their souls and his death was on the great tragedies because of his death and the death of Othman before him were the cause of this great fitnah to this Nation and their killers are the worst of the creatures in allah creations."
End of his quote.
And then he continued saying:
When hussien was killed on the day of Ashura, The oppressive, unjust people killed him and allah has bless Hussien RA with the Shahada(martyr) just like he blesses Ahlul Al Bayit, Allah blessed Hamza, Jaafir, His father Ali, and others. and his martyr which increased his level. as he and his brother(Hasan RA) are the leaders of the youth in Pradise. and High levels can't be reached without hardship. as the prophet Mohamed (PBUH) was asked: "O Messenger of Allah(s.a.w)! Which of the people is tried most severely?" He said: "The Prophets, then those nearest to them, then those nearest to them. A man is tried according to his religion; if he is firm in his religion, then his trials are more severe, and if he is frail in his religion, then he is tried according to the strength of his religion. The servant shall continue to be tried until he is left walking upon the earth without any sins." (Hasan)
So Hasan and Hussien (RA) they have already reached that a high level from Allah (SWT) and they have never witnessed hardships before as what happened to their good ancestors. They were born in the best tome of Islam, Raised in the best time of Isalm, and Muslims honours them, and the prophet Mohamed (PBUH) died and they haven’t reached the age where they can decide correct from false . So it was a bless from Allah that he gave them this great trial that didn't happen to their ancestors from Ahlul Al Bayt and he already had trailed who is better than them, "Ali bin Abi Talib (RA)" and indeed Ali(RA) is way better than them. and he was Martyred too. and with the death of Hussien. Problems have risen between Muslims. Just like how the death of Othman has caused a great problem between Muslims and because of it the Ummah has been divided until this day.
So when hussien went out of Madinah and has witnessed that things have changed(People in Kufah betrayed him) He asked those who encountered him either to go back, Go to another city, or follow his cousin Yazid but they didn't allow him unless they take him as a prisoner so hey fought them and killed him unjustly along with his people all of them are considered Martyr,Which is a bless from Allah and because of it he followed the Ahlul Al Bayt the good and the pure people. and Allah by this bless has disgraced those who killed him"
End of the quote from "Ibn Taiymiah"
Ibn Kathir says in his book (11/579-580) "البداية والنهاية":
It is a must for every Muslim to be sad because of this disaster(referring to death of Hussien) he is the leader of Muslims. The most greatest companion and the son of Fatima RA the daughter of the prophet Mohamed (PBUH) and he was a pious, brave,and generous. But what others do to show that they were sad because of his death such as trying to be extremely sad which is sometimes considered artificial and hypocrisy. If they were really sad for his death they should have felt more sad for the death of Ali (RA) who was much greater than Hasan and Hussien (RA) and he was killed on Friday while going to pray the Fajir in the Mosque on the 17th of Ramadan.
Source
Hassan Al Basri says:
After Umar bin Sa'ad had killed Hussien(RA) and cut his head. He came to Ibn Ziyad (who was the ruler of Kufah under Yazid's regime) saying:
"I have killed the one who has the best father and mother"
So Ibn Ziyad got angry at Umar and said if you knew his status why did you kill him ? He continued saying I didn't order you to kill him. At that time he ordered the soldiers to cut Umar's head and other narrations say that Yazid who has ordered the soldiers to cut Umar's head not Ibn Ziyad
Source
OFF-TOPIC:
Ibn Kathir says in his book (12/Page 45) "البداية والنهاية"
Ibn Ziyad was attacked by Muslim tribes in North of Iraq by a Muslim man called: Ibrahim Ibn al Ashtar who left kuffa and travelled to the north to kill Ibn Yazid , after he reached their, He gathered all the tribes and then he moved to where Ibn Yazid and his army were staying . After the two armies have met. Ibrahim Ibn al Ashtar encouraged the tribes to go and battle by saying:
This is the man who killed the son of the daughter of the prophet Mohammed(PBUH), He has done what Pharaoh didn't do to sons of Jerusalem. he didn't even allow Hussien to drink water. neither he allowed the children or the wives. Kill him and kill his soldiers and get rid of your anger by throwing your spears to their chest.
For more info about what happened in the battle clock on the source
At the end of the battle Ibrahim Ibn al Ashtar killed Ibn Ziad
