Short answer
Al-Hasan and al-Husayn were the sons of Ali ibn Abi Talib and Fatimah, the grandsons of the Prophet ﷺ. He loved them dearly and called them the leaders of the youth of Paradise. Al-Hasan made peace among the Muslims in 41 AH, as he had foretold, and al-Husayn was martyred at Karbala in 61 AH.
Sunni Muslims love and honour al-Hasan and al-Husayn as members of the Ahl al-Bayt and Companions of the Prophet ﷺ. Sunni scholars praise al-Hasan's peace with Mu'awiya as the fulfilment of a prophecy.
The four schools agree on this.
Explanation
Al-Hasan was born in Madinah in 3 AH and al-Husayn about a year later, in 4 AH. The Prophet ﷺ named them, carried them on his shoulders, let them climb on his back in prostration, and came down from the pulpit to pick them up when they stumbled. He said they were his two sweet basils in this world, and the leaders of the youth of Paradise.
Al-Hasan was known for his gentleness and generosity. After his father Ali was killed in 40 AH, the people of Iraq pledged to al-Hasan. Rather than fight a long war between Muslims, he made peace with Mu'awiya in 41 AH and handed over rule. That year became known as the year of unity. Years earlier the Prophet ﷺ had said from the pulpit: "This son of mine is a sayyid, and perhaps Allah will make peace through him between two great groups of Muslims." He died in Madinah around 49 to 50 AH and is buried in al-Baqi.
Al-Husayn lived in Madinah during Mu'awiya's rule. When Yazid took power in 60 AH, al-Husayn refused to pledge, set out for Kufa at the invitation of its people, and was killed at Karbala on 10 Muharram 61 AH. See What happened at Karbala?
The Prophet's ﷺ descendants through Fatimah come through these two brothers; their descendants are known as sharifs and sayyids. See Who are the Ahl al-Bayt? and Who was Fatimah?
The evidence
Quran, Surah Al-Ahzaab 33:33
وَقَرْنَ فِى بُيُوتِكُنَّ وَلَا تَبَرَّجْنَ تَبَرُّجَ ٱلْجَٰهِلِيَّةِ ٱلْأُولَىٰ ۖ وَأَقِمْنَ ٱلصَّلَوٰةَ وَءَاتِينَ ٱلزَّكَوٰةَ وَأَطِعْنَ ٱللَّهَ وَرَسُولَهُۥٓ ۚ إِنَّمَا يُرِيدُ ٱللَّهُ لِيُذْهِبَ عَنكُمُ ٱلرِّجْسَ أَهْلَ ٱلْبَيْتِ وَيُطَهِّرَكُمْ تَطْهِيرًۭا ﴿٣٣﴾
And abide in your houses and do not display yourselves as [was] the display of the former times of ignorance. And establish prayer and give zakah and obey Allah and His Messenger. Allah intends only to remove from you the impurity [of sin], O people of the [Prophet's] household, and to purify you with [extensive] purification.
The verse of the household, which the hadith of the cloak links to them.
Sahih al-Bukhari 2704
Sahih... إن ابني هذا سيد ولعل الله أن يصلح به بين فئتين عظيمتين ...
"This son of mine is a sayyid, and perhaps Allah will make peace through him between two great groups."
The prophecy of al-Hasan's peace.
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 3753
Sahih... هما ريحانتاي من الدنيا ...
"They are my two sweet basils in this world."
His love for them.
Where the ruling is recorded
- al-Isaba fi Tamyiz al-Sahaba (Ibn Hajar), Entries on al-Hasan ibn Ali and al-Husayn ibn Ali. Their births, lives and deaths.
- Fath al-Bari (Ibn Hajar), Commentary on Kitab al-Sulh, chapter on the Prophet's words about al-Hasan. Explains the peace of 41 AH.
- al-Bidaya wa al-Nihaya (Ibn Kathir), Events of the year 41 AH, the year of unity. Al-Hasan's handing over of rule.
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.