Short answer
On 10 Muharram 61 AH (October 680 CE), al-Husayn ibn Ali, the Prophet's ﷺ grandson, was killed at Karbala in Iraq with members of his family and companions by an army sent by the Umayyad governor of Kufa. Sunni Muslims regard him as a martyr and his killing as a great crime.
Sunni scholars hold that al-Husayn was killed unjustly as a martyr, that those who killed him and ordered it committed a grave sin, and that Muslims should love the Ahl al-Bayt, remember them with respect, and avoid both excess about them and disrespect toward any of the Companions.
The four schools agree on this.
Explanation
After Mu'awiya died in 60 AH, his son Yazid took power. Al-Husayn, then in Makkah, refused to pledge allegiance. People of Kufa wrote many letters inviting him to come and lead them. He sent his cousin Muslim ibn Aqil ahead, then set out with his family. Several senior Companions, including Ibn Abbas and Ibn Umar, advised him not to go, fearing the people of Kufa would abandon him.
Meanwhile the governor Ubaydullah ibn Ziyad took control of Kufa and had Muslim ibn Aqil killed. An army under Umar ibn Sa'd stopped al-Husayn's small group at Karbala, by the Euphrates, and cut them off from the water. On the day of Ashura, 10 Muharram 61 AH, al-Husayn and around seventy men of his family and companions were killed after a brave stand. The women and children, including Zaynab bint Ali and his son Ali Zayn al-Abidin, were taken to Kufa and then Damascus. Anas ibn Malik reports in Bukhari that al-Husayn's head was brought to Ubaydullah ibn Ziyad.
Sunni Muslims grieve for al-Husayn. The Prophet ﷺ called al-Hasan and al-Husayn his two sweet basils in this world and the leaders of the youth of Paradise. Sunni scholars condemn the killing. Many of them, including Hanafi scholars, hold back from cursing Yazid by name while judging his rule harshly; others allowed it. They agree the crime was grave.
The tenth of Muharram was already a day of fasting in the Sunnah before Karbala. Sunni Muslims keep that fast and remember al-Husayn with love and dua. See Ashura and Who were Hasan and Husayn?
The evidence
Sahih al-Bukhari 3753
Sahih... هما ريحانتاي من الدنيا ...
"They are my two sweet basils in this world."
His love for al-Hasan and al-Husayn.
Jami' at-Tirmidhi 3768
Grading to be checked... الحسن والحسين سيدا شباب أهل الجنة ...
"Al-Hasan and al-Husayn are the leaders of the youth of Paradise."
Their rank, graded sahih.
Sahih al-Bukhari 3748
Sahih... أتي عبيد الله بن زياد برأس الحسين ...
"The head of al-Husayn was brought to Ubaydullah ibn Ziyad."
Anas witnessed the aftermath of Karbala.
Where the ruling is recorded
- al-Bidaya wa al-Nihaya (Ibn Kathir), Events of the year 61 AH, the killing of al-Husayn. Ibn Kathir's detailed account, with his judgement on the reports.
- al-Isaba fi Tamyiz al-Sahaba (Ibn Hajar), Entry on al-Husayn ibn Ali. His life and death.
- Fath al-Bari (Ibn Hajar), Commentary on Kitab Fada'il Ashab al-Nabi, chapter on the virtues of al-Hasan and al-Husayn. Explains the hadith of Anas about the head.
This answer is pending review by the scholars on our team. Quran translation by Sahih International. Hadith Arabic from the standard editions of the six books and the Muwatta, with gradings as recorded there; the English is our own rendering. Spotted a mistake? Tell us.