Short answer
Fatimah al-Zahra was the youngest daughter of the Prophet ﷺ and Khadijah, the wife of Ali ibn Abi Talib and the mother of Hasan and Husayn. He called her a part of him and the leader of the women of Paradise.
Sunni Muslims honour Fatimah as the most beloved daughter of the Prophet ﷺ, one of the best women of all time, and the mother through whom his descendants come. Love for her is part of love for the Ahl al-Bayt.
The four schools agree on this.
Explanation
Fatimah bint Muhammad was born in Makkah to the Prophet ﷺ and Khadijah, most reports say a few years before revelation began. She grew up through the hardships of Makkah, the boycott and the death of her mother. She was known as al-Zahra (the radiant) and for her likeness to her father in the way she walked and spoke.
In Madinah she married Ali ibn Abi Talib, around 2 AH. Reports differ on her exact age at the time. Their home was simple and poor. When she asked her father for a servant, he taught her and Ali to say subhanAllah, alhamdulillah and Allahu akbar before sleep instead. Their children were Hasan, Husayn, Zaynab and Umm Kulthum. All the Prophet's ﷺ descendants come through her.
He loved her deeply. He stood up for her when she came in and seated her in his place. He said she was a part of him, and whoever angered her angered him. In his final illness he told her she would be the leader of the women of Paradise, and that she would be the first of his family to join him.
She died about six months after him. See How did Fatimah die?, Who are the Ahl al-Bayt? and the family tree.
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 about the Prophet's household, which the hadith of the cloak links to Fatimah and her family.
Sahih al-Bukhari 3714
Sahih... فاطمة بضعة مني فمن أغضبها أغضبني ...
"Fatimah is a part of me, so whoever angers her angers me."
Her place with her father.
Sahih al-Bukhari 3624
Sahih... سيدة نساء أهل الجنة ...
"The leader of the women of the people of Paradise."
His glad tidings to her in his last illness.
Where the ruling is recorded
- al-Isaba fi Tamyiz al-Sahaba (Ibn Hajar), Entry on Fatimah al-Zahra. Her birth, marriage, children and death.
- Fath al-Bari (Ibn Hajar), Commentary on Kitab Fada'il Ashab al-Nabi, chapter on the virtues of Fatimah. Explains the hadith of her virtues.
- al-Bidaya wa al-Nihaya (Ibn Kathir), Events of the year 2 AH, marriage of Ali and Fatimah. Dates the marriage after Badr.
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.