Short answer
The Mahdi is a righteous man from the family of the Prophet ﷺ, from the descendants of Fatimah, who will appear near the end of time and fill the earth with justice. Sunni Muslims believe in him on the basis of authentic hadith. He is not a prophet, and Isa (Jesus) will return in his time.
Belief in the coming of the Mahdi is part of Sunni creed, based on many hadith that together are authentic in meaning, and Hanafi creed commentaries such as Mulla Ali al-Qari's list him among the signs of the Hour. He brings no new revelation and follows the shariah of the Prophet ﷺ.
The four schools agree on this.
Explanation
"Mahdi" means "the rightly guided one". The hadith describe him as a man from the Prophet's household, from the children of Fatimah, whose name will match the Prophet's name and whose father's name will match the name of the Prophet's father: Muhammad ibn Abdullah. He will rule with justice after the earth has been filled with injustice and oppression.
These hadith are not in al-Bukhari or Muslim by name. They are in Abu Dawud, al-Tirmidhi, Ibn Majah and other collections, and many of them are graded sahih or hasan. Because so many narrations from different Companions say the same thing, most Sunni scholars hold that the coming of the Mahdi is established beyond doubt, even if details in individual reports are weak.
Sunni and Shia belief differ here. Shia Muslims believe the Mahdi is their twelfth imam, born in the ninth century and now in hiding. Sunni scholars believe the Mahdi will be born and grow up in his own time, like anyone else, and will be recognised by the events around him.
History has seen many people claim to be the Mahdi, and some still do. The true Mahdi does not need to advertise himself, and no Muslim is asked to search for him or follow a claimant. Your duty is to hold to the Quran and Sunnah. The Mahdi's time is linked to the coming of the Dajjal and the return of Isa, covered in Imam Mahdi and signs of the Day of Judgement.
The evidence
Sunan Abi Dawud 4282
Graded hasan sahih... يُوَاطِئُ اسْمُهُ اسْمِي وَاسْمُ أَبِيهِ اسْمَ أَبِي ...
"His name will match my name, and his father's name will match my father's name."
Describes the Mahdi as a man from the Prophet's family named Muhammad ibn Abdullah.
Sunan Abi Dawud 4284
Graded sahih... الْمَهْدِيُّ مِنْ عِتْرَتِي مِنْ وَلَدِ فَاطِمَةَ ...
"The Mahdi is from my family, from the descendants of Fatimah."
States the Mahdi's lineage through Fatimah.
Where the ruling is recorded
- Sharh al-Fiqh al-Akbar (Mulla Ali al-Qari), Section on the signs of the Hour. Lists the coming of the Mahdi among the signs of the Hour.
- Sunan Abi Dawud, Kitab al-Mahdi. Abu Dawud gathered the hadith on the Mahdi in a book of their own.
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.