Virtues of Ahl Al-Bayt
Question

What is the virtue of Ahl al-Bayt and their superiority over other people? Will they intercede for the people on the Day of Resurrection?
Praise be to Allah.Who are Ahl Al-Bayt?
In the answer to question no. 10055 , we explained that Ahl Al-Bayt are the members of the household of the Prophet (peace and blessings of Allah be upon him). At the end of that answer we said: 
“So the family members of the Prophet (peace and blessings of Allah be upon him) are: his wives, his children, and Banu Hashim, Banu `Abd Al-Muttalib and their freed slaves.” 
Rights of Ahl Al-Bayt
Allah has bestowed upon the Ahl al-Bayt of the Prophet (peace and blessings of Allah be upon him) numerous virtues. Ahl As-Sunnah wa’l-Jama`ah are unanimously agreed that it is obligatory to love them and respect their rights.  
Shaykh Al-Islam Ibn Taymiyah (may Allah have mercy on him) said: 
“Similarly the family of the Messenger of Allah (peace and blessings of Allah be upon him) have rights which must be paid attention to. Allah has allocated to them a share of the khums and fay’, and He has enjoined that blessings be sent upon them along with the Messenger of Allah (peace and blessings of Allah be upon him), who said to us: “Say: ‘Allahumma salli ‘ala Muhammadin wa ‘ala ali Muhammadin kama salayta ‘ala ali Ibrahim, innaka Hamidun Majid, wa barik ‘ala Muhammadin wa ‘ala ali Muhammadin kama barakta ‘ala ali Ibrahim innaka Hamidun Majid (O Allah, send prayer upon Muhammad and upon the family of Muhammad, as You sent prayers upon the family of Ibrahim; You are indeed Worthy of Praise, Full of Glory. And send blessings upon Muhammad and upon the family of Muhammad, as You sent blessings upon the family of Ibrahim; You are indeed Worthy of Praise, Full of Glory)’”. (Majmu’ Al-Fatawa, 3/407) 
And he also said: 
“Similarly, with regard to the family of the Messenger of Allah (peace and blessings of Allah be upon him) (Ahl Al-Bayt), it is obligatory to love them, be loyal to them and respect their rights.” (Majmu’ Al-Fatawa, 28/491)
Virtues of Ahl Al-Bayt
The virtues of the family of the Prophet (peace and blessings of Allah be upon him) are: 

Allah says (interpretation of the meaning):

“O wives of the Prophet! You are not like any other women. If you keep your duty (to Allah), then be not soft in speech, lest he in whose heart is a disease (of hypocrisy, or evil desire for adultery) should be moved with desire, but speak in an honourable manner. And stay in your houses, and do not display yourselves like that of the times of ignorance, And perform As-Salah (Iqamat-As-Salah), and give Zakah and obey Allah and His Messenger. Allah wishes only to remove Ar-Rijs (evil deeds and sins) from you, O members of the family (of the Prophet), and to purify you with a thorough purification.” [Al-Ahzab 33:32, 33]
This does not apply only to the Prophet’s wives (may Allah be pleased with them); rather it is narrated in the sahih Sunnah that it also includes others. It was narrated that ’Aishah (may Allah be pleased with her) said:
The Prophet (peace and blessings of Allah be upon him) went out one morning wearing a striped cloak of black camel hair. Al-Hasan ibn ‘Ali came and he enfolded him in the cloak, then al-Husayn came and he enfolded him in it, then Fatimah came and he enfolded her in it, then ‘Ali came and he enfolded him in it, then he said: “Allah wishes only to remove Ar-Rijs (evil deeds and sins) from you, O members of the family (of the Prophet), and to purify you with a thorough purification.” [Al-Ahzab 33:33]   (Narrated by Muslim, 2424) 

Allah says (interpretation of the meaning):

“The Prophet is closer to the believers than their own selves, and his wives are their (believers’) mothers (as regards respect and marriage).” [Al-Ahzab 33:6]

It was narrated that Wathilah ibn Al-Asqa’ said: I heard the Messenger of Allah (peace and blessings of Allah be upon him) say:

“Allah, may He be glorified and exalted, chose Kinanah from among the children of Isma’il (peace be upon him), and He chose Quraysh from among Kinanah, and He chose Banu Hashim from among Quraysh, and He chose me from among Banu Hashim.” (Narrated by Muslim, 2276) 

It was narrated that Zayd ibn Arqam said: The Messenger of Allah (peace and blessings of Allah be upon him) stood and addressed us at a watering place called Khumm, between Makkah and Madinah. He praised and glorified Allah, and he exhorted and reminded us, then he said: “O people, I am only human, and soon the messenger of my Lord will come to me and I will respond. I am leaving among you two weighty things, the first of which is the Book of Allah in which is guidance and light. Follow the Book of Allah and hold fast to it.” And he encouraged us to adhere to the Book of Allah, then he said: “And the people of my household, I remind you of Allah with regard to the people of my household, I remind you of Allah with regard to the people of my household, I remind you of Allah with regard to the people of my household.” (Narrated by Muslim, 2408) 

The Companions of the Prophet (peace and blessings of Allah be upon him) paid attention to this commandment; foremost among them were Abu Bakr Al-Siddiq and `Umar ibn Al-Khattab (may Allah be pleased with them). 
Al-Bukhari (3508) and Muslim (1759) narrated that Abu Bakr (may Allah be pleased with him) said to ‘Ali (may Allah be pleased with him): “By the One in Whose Hand is my soul, kinship with the Messenger of Allah (peace and blessings of Allah be upon him) is dearer to me than kinship with my own people.” 
Al-Bukhari also narrated in his Sahih (3509) that Abu Bakr (may Allah be pleased with him) said: “Pay attention to the rights of the household of Muhammad (peace and blessings of Allah be upon him).” 
Al-Hafiz Ibn Hajar (may Allah have mercy on him) said:
“The words “Pay attention to the rights of the household of Muhammad (peace and blessings of Allah be upon him)” are addressed to the people, enjoining them to fulfill the rights of Ahl Al-Bayt.” (Fath Al-Bari, 7/79) 
With regard to the respect of `Umar ibn Al-Khattaab for the relatives of the Prophet (peace and blessings of Allah be upon him), that is apparent from many things, such as his preferring to put the ahead of himself or any other people with regard to allocating shares from bayt al-mal. 
Shaykh Al-Islam Ibn Taymiyah (may Allah have mercy on him) said: 
“Moreover, when ‘Umar ibn Al-Khattab (may Allah be pleased with him) set up the registry of stipends, he wrote  down people’s names according to their lineage, so he started with those who were closest in lineage to the Messenger of Allah (peace and blessings of Allah be upon him), When he had finished listing the Arabs, he listed the non-Arabs. This is how the registry was at the time of the Rightly-Guided Caliphs, and all the caliphs of the Umayyad and ‘Abbasid dynasties, until things changed after that.”  (Iqtida’ Al-Sirat Al-Mustaqim, p. 159, 160) 
Will Ahl Al-Bayt intercede on the Day of Resurrection? 
There is no intercession specifically for the family of the Prophet (peace and blessings of Allah be upon him), rather that is something that may be granted to all those among the righteous, the martyrs and the scholars with whose intercession Allah is pleased, whether they are among Ahl Al-Bayt or are from among the ordinary people. 
In the answer to question no. 21672 , we said: “Intercession for those who committed sin will not be granted only to the Prophet (peace and blessings of Allah be upon him), rather the prophets, the martyrs, the scholars, the righteous and the angels will also join in that. A man’s righteous deeds may also intercede for him. But the Prophet (peace and blessings of Allah be upon him) will have the greatest share of intercession.”
Thus we may know the response to the extremists among the Rafidis who claim that intercession is only for the family of the Prophet (peace and blessings of Allah be upon him). Rather it is stated in their books that Ahl Al-Bayt are the ones who will admit people to Paradise and to Hell! This is just part of a lengthy list of the different types of exaggeration that they have, the source of which is their ignorance of the religion of Allah, may He be exalted, and their keeping away from the texts of Revelation in the Quran and Sunnah. 
We advise you to read an essay entitled Fadl Ahl Al-Bayt wa ‘Uluw Makanatihim ‘inda Ahl Al-Sunnah wal-Jama`ah, written by Shaykh ‘Abd Al-Muhsin ibn Hamad Al-‘Abbad Al-Badr, which we have benefited from in composing this answer, and which contains additional research on this topic. It is short but very useful, and may be seen here [in Arabic].
For more, please see these answers: 21981 and 112754 . 
And Allah knows best.