Narrated Jabir ibn Samura:  I heard the Prophet (s) saying:

"There will be Twelve Commanders." He then said a sentence which I did not hear. My father said, the Prophet added, "All of them will be from Quraysh."
[Sahih al-Bukhari (English),  Hadith: 9.329, Kitabul Ahkam;
Sahih al-Bukhari, (Arabic), 4:165, Kitabul Ahkam]

The Prophet (s) said:

"The Religion (Islam) will continue until the Hour (Day of Resurrection), having Twelve Caliphs for you, all of them will be from Quraysh."
[Sahih Muslim, (English), Chapter DCCLIV, v3, p1010, Tradition #4483; Sahih Muslim (Arabic), Kitab al-Imaara, 1980 Saudi Arabian Edition, v3, p1453, Tradition #10]

What the Sunni Scholars say:

Ibn al-'Arabi:
We have counted the Amirs after the Holy Prophet (s.a.w.s.) as twelve.  We found them as follows: Abu Bakr, 'Umar, 'Uthman, Ali, Hasan, Mu'awiyah, Yazid, Mu'awiyah ibn Yazid, Marwan, 'Abd al-Malik ibn Marwan, Yazid bin 'Abd al-Malik, Marwan bin Muhammad bin Marwan, As-Saffah... After this there were twenty-seven caliphs from the Bani Abbas.

Now if we consider twelve of them we can reach only till Sulayman. If we take the literal meaning we have only five of them and to these we add the four Righteous Caliphs, and 'Umar bin 'Abd al-'Aziz... 
I cannot understand the meaning of this Hadith.
[Ibn al-'Arabi, Sharh Sunan Tirmidhi, 9:68-69]

Qadi 'Iyad al-Yahsubi:
The number of Caliphs are more than that. To limit their number to twelve is incorrect. The Holy Prophet (s.a.w.s.) did not say that there will be only twelve and there is no scope for more. Hence it is possible that there can be more.
[Al-Nawawi, Sharh Sahih Muslim, 12:201-202; Ibn Hajar al-'Asqalani, Fath al-Bari, 16:339]

Jalal al-Din al-Suyuti:
There are only twelve Caliphs until the Day of Judgement. And they will continue to act on truth, even if they are not continuous.

We see that from the twelve, four are the Righteous Caliphs, then Hasan, then Mu'awiyah, then Ibn Zubayr, and finally 'Umar bin 'Abd al-'Aziz. They are eight. Four of them remain. Maybe Mahdi, the Abbasid could be included as he is an Abbasid like 'Umar bin 'Abd al-'Aziz was an Umayyad. And Tahir 'Abbasi will also be included because he was a just ruler. Thus two more are yet to come. One of them is Mahdi, because he is from the Ahlul Bayt (a.s.).
[Al-Suyuti, Tarikh al-Khulafa, Page 12; Ibn Hajar al-Haytami, Al-Sawa'iq al-Muhriqa Page 19]

Ibn Hajar al-'Asqalani:
No one has much knowledge about this particular hadith of Sahih Bukhari.

It is not correct to say that these Imams will be present at one and the same time.
[Ibn Hajar al-'Asqalani, Fath al-Bari 16:338-341]

Ibn al-Jawzi:
The first Caliph of Bani Umayya was Yazid ibn Mu'awiyah and the last, Marwan Al-Himar. Their total is thirteen. 'Uthman, Mu'awiyah and ibn Zubayr are not included as they were among the Companions of the Holy Prophet (s).
If we exclude Marwan bin al-Hakam because of the controversy about his being a Companion or that he was in power even though Abdullah ibn Zubayr had the support of the people. Then we can get the figure of Twelve. … When the Caliphate came out of the Bani Umayya, a great disturbance arose. Until the Bani Abbas established themselves. Hence, the original conditions had changed completely.
[Ibn al-Jawzi, Kashf al-Mushkil, as quoted in Ibn Hajar al-'Asqalani, Fath al-Bari 16:340 from Sibt Ibn al-Jawzi]

Al-Nawawi:
It could also mean that the twelve Imams will remain during the period of Islam's supremacy. The time when Islam will be a dominant religion. These Caliphs will, during their tenure, glorify the religion.
[Al-Nawawi, Sharh Sahih Muslim ,12:202-203]

Al-Bayhaqi:
This number (twelve) is found till the period of Walid ibn 'Abd al-Malik. After this, there was chaos and disturbance. Then came the Abbasid dynasty. This report has increased the number of Imams. If we neglect some of their characteristics which came after the disturbance, then their number will be much higher."
[Ibn Kathir, Ta'rikh, 6:249; Al-Suyuti, Tarikh al-Khulafa Page 11]

Ibn Kathir:
Whosoever follows Bayhaqi and agrees with his assertion that Jama'ah means those Caliphs who came intermittently till the time of Walid ibn Yazid ibn 'Abd al-Malik the transgressor comes under the purview of the tradition quoted by us criticising and denouncing such people.

And if we accept the Caliphate of Ibn Zubayr before 'Abd al-Malik the total shall be sixteen. Whereas their total should be twelve before 'Umar ibn 'Abd al-'Aziz. In this method Yazid ibn Mu'awiyah will be included and not 'Umar ibn 'Abd al-'Aziz. However, it is established that the majority of the 'ulama accept 'Umar ibn 'Abd al-'Aziz as a truthful and a just Caliph.
[Ibn Kathir, Ta'rikh, 6:249-250]

Confused?

We need another Sunni scholar to finally clarify who these Twelve Successors, Caliphs, Amirs and Imams really are:

The famous scholar al-Dhahabi says in Tadhkirat al-Huffaz, vol. 4, p. 298, and Ibn Hajar al-'Asqalani says in al-Durar al-Kaminah, vol. 1, p. 67 that Sadruddin Ibrahim bin Muhammad bin al-Hamawayh al-Juwayni al-Shafi'i was a great scholar of Hadith.  The same Al-Juwayni reports from Abdullah ibn Abbas (r) from the Prophet (s) who 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 also narrates from Ibn 'Abbas (r) from the Prophet (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) 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) 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 that the Messenger of Allah (s) informed: "I and Ali and Hasan and Husayn and nine of the descendants of Husayn are the purified ones and the inerrant."
[Al-Juwayni, Fara'id al-Simtayn, Mu'assassat al-Mahmudi li-Taba'ah, Beirut 1978, p. 160.]

Amongst all the Islamic schools of thought, only the Shi'ah Imamiyyah Ithna 'Ashariyyah (Twelver Shi'ites) believe in these individuals as the Twelve rightful successors of the Prophet (s) and obtain their understanding of Islam from them.