Al-Juwayni [1] reports that Abdullah ibn Abbas remarked that the Messenger of Allah (s.a.w.s.) said, "I am the chief of the Prophets and Ali ibn Abi Talib is the chief of successors, and after me my successors shall be twelve, the first of them being Ali ibn Abi Talib and the last of them being Al Mahdi."
Al-Juwayni has also mentioned another tradition from Ibn 'Abbas (r.a.) that he narrates from the Messenger of Allah (s.a.w.s.): "Certainly my Caliphs and my legatees and the Proofs of Allah upon his creatures after me are twelve. The first of them is my brother and the last of them is my (grand) son."
He was asked: "O Messenger of Allah, who is your brother?"
He said, "Ali ibn Abi Talib"
Then they asked, "And who is your son?"
The Holy Prophet (s.a.w.s.) replied, "Al Mahdi, the one who will fill the earth with justice and equity like it would be brimming with injustice and tyranny. And by the One Who has raised me as a warner and a give of good tidings, even if a day remains for the life of this world, the Almighty Allah will prolong this day to an extent till he sends my son Mahdi, then he will make Ruhullah 'Isa ibn Maryam (a.s.) to descend and pray behind him (Mahdi). And the earth will be illuminated by his radiance. And his power will reach to the east and the west."
Al-Juwayni also narrates from his chain of narrators that the Messenger of Allah (s.a.w.s.) informed: "I and Ali and Hasan and Husayn and nine of the descendants of Husayn are the purified ones and the infallible." [2]
Among the scholars of the School of the Caliphate it was the general tendency, due to the political expediency, to conceal such traditions from the people. A majority of their scholars have tried their best to explain away these traditions in a confusing manner. They have endeavoured to make vague guesses regarding the names of the Caliphs mentioned in these reports. On the other hand the Holy Prophet (s.a.w.s.) has mentioned clearly, by name, his Twelve Successors.
It is not possible to relate all such traditions in this brief treatise. However we shall present the biographical sketches of the Imams (a.s.) who have been named by the Messenger of Allah (s.a.w.s.).

An Important Advice
One of the points in the traditions mentioned above is that each of the twelve will be from the Quraysh. After them there will be chaos. Another point is that the religion will remain established till the twelve Caliphs of Quraysh are present. When they die, the earth will swallow its inhabitants.
The above two points thus prove that after the twelve Caliphs of the Holy Prophet (s.a.w.s.) the earth will be destroyed.
Therefore it is necessary that one of these twelve successors should have such a long life that it should surpass the age of this earth.
This is exactly how it happened. The Twelfth successor of the Holy Prophet (s.a.w.s.) was bestowed with a long life. He is Al-Mahdi Muhammad ibn Hasan Al Askari (a.s.).
The traditions discussed in this treatise prove the Imamat of these Twelve Noble Personalities. Not anyone else.

Notes:
[1] Al-Dhahabi says in Tadhkirat al-Huffaz, vol. 4, p. 298, that Sadruddin Ibrahim bin Muhammad bin al-Hamawayh al-Juwayni al-Shafi'i was a great scholar of Hadith. Also see his biographical note in Ibn Hajar al-'Asqalani, al-Durar al-kaminah, vol. 1, p. 67.
[2] Al-Juwayni, Fara'id al-Simtayn, pg 160.