Short answer
Fatimah died of illness in Madinah about six months after the Prophet ﷺ, in 11 AH, while still young, and Ali buried her at night. In his last illness her father had told her she would be the first of his family to follow him.
The Sunni historical account, from Sahih al-Bukhari, is that Fatimah lived about six months after her father and then died, and that Ali buried her at night. Sunni sources describe an illness and grief, not an injury.
The four schools agree on this.
Explanation
In his final illness, the Prophet ﷺ whispered to Fatimah and she wept. He whispered again and she laughed. After his death she explained: first he told her he would die of that illness, then he told her she would be the first of his family to follow him, and that she would be the leader of the women of Paradise.
It happened as he said. Aisha reports in Sahih al-Bukhari that Fatimah lived six months after him. That places her death around Ramadan of 11 AH (632 CE). She was in her late twenties according to many reports. Her husband Ali buried her at night and prayed over her. She is buried in al-Baqi in Madinah, according to the best-known view.
The Sunni sources describe her falling ill and weakening, deeply grieved by the loss of her father. Some Shia accounts say she died from an injury. Sunni hadith scholars and historians do not accept those reports, since they have no sound chains. Muslims of all backgrounds share love and grief for her.
The same hadith in Bukhari tells that Ali gave his public pledge to Abu Bakr after her death. Read Who was Fatimah in Islam? and How was Abu Bakr chosen as the first caliph?
The evidence
Sahih al-Bukhari 4240
Sahih... وعاشت بعد النبي صلى الله عليه وسلم ستة أشهر ...
"She lived six months after the Prophet."
Dates her death and says Ali buried her at night.
Sahih al-Bukhari 3626
Sahih... أول أهل بيته يتبعه فضحكت ...
"That I would be the first of his family to follow him, so I laughed."
His prophecy to her, which came true.
Where the ruling is recorded
- al-Isaba fi Tamyiz al-Sahaba (Ibn Hajar), Entry on Fatimah al-Zahra. Her death six months after the Prophet and her burial.
- Fath al-Bari (Ibn Hajar), Commentary on Kitab al-Maghazi, chapter on the expedition of Khaybar, the hadith of Fatimah and Abu Bakr. Explains the report of her death and Ali's pledge.
- al-Bidaya wa al-Nihaya (Ibn Kathir), Events of the year 11 AH, death of Fatimah. Records the reports of the month of her death and her burial place.
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.