The term Ahlul-Bayt has been mentioned twice in the Quran.  Once for the household of Prophet Ibrahim (pbuh) and the other in the verse we will be addressing, which is verse 33 of surah Ahzab. In the end of this verse, Allah (swt) speaks of His decision to purify the Ahlul-Bayt. This verse has been analyzed from different perspectives by Islamic scholars throughout history; one of those perspectives being who this verse is exactly speaking of. There are several theories on this subject, one of them belongs to Shia scholars who based on the many hadiths that have touched on this matter, believe that the term is referring to the prophet (pbuh), Imam Ali (as), Lady Zahra (as), and Imams Hasan and Husein (as)  and the infallible descendants of Imam Husein (as).  As for the Sunni school of thought and its view on the issue, there are several theories:
1- Only the prophets (pbuh) wives are meant.
2- The prophets (pbuh) wives, Imam Ali (as), Lady Zahra (as), and Imams Hasan and Husein (as).
3- Those who are considered part of his household, let it be his wives, relatives, or close ones (meaning the family of Abbas, the family of Aqil, the family of Jafar and the family of Ali).
4- The residents of Beytul-Haram (in Mekkah).
5- Those belonging to the prophets mosque. 
The latter three theories dont have many supporters and arent spoken of much. The reasoning for the first theory is a saying by Akramah. He has been narrated shouting that the only persons meant by Ahlul-Bayt are the prophets (pbuh) wives.  This theory hasnt been very well received by Sunni commentators of the Quran  , add to that the fact that this is only the saying of Akramah, or him repeating what he had heard from Ibn Abbas, and it in no way is connected to the prophet (pbuh) and there are no narrations of him (the prophet) saying such a thing.
Almost all Sunni tafsirs have listed hadiths that say that the Ahlul-Bayt are indeed the prophet (pbuh), Imam Ali (as), Lady Zahra, and Imams Hasan and Husein (as). They narrate the famous hadith by Umm Salmah known as the Hadith of Kisa (cloak). Therefore all of them consider these five as at least part of the Ahlul-Bayt. Some of the tafsirs which have cited these hadiths or the messages of these hadiths are as follows:
1- The tafsir of Fakhr Razi (Mafatihul-Gheib), vol. 25, pg. 168, in the part where the Verse of Tathir (purification) is discussed.
2- The tafsir of Ibn Abi Hatam (Tafsirul-Quran al-Adhim), vol. 9, pg. 3132, under the discussion on the Verse of Purification.
3- The tafsir of Nidamuddin Neyshabouri (Tafsir Gharaibul-Quran wa Raghaibul-Furqan), vol. 5, pg. 460, Verse of Purification.
4- The tafsir of Muhammad ibn Jurair Tabari (Jameul-Bayan fi Tafsiril-Quran), vol. 22, pg. 5, Verse of Purification.
5- The tafsir of Haskani, (Shawahidul-Tanzil li Qawaidil Tafdil), vol. 2, pg. 18. In his discussion on the Verse of Purification, he has gathered the related hadiths very well.
6- The tafsir of Suyuti (Al-Durrul-Manthur fi Tafsiril-Mathur), vol. 5, pg. 198, under the Verse of Purification.
7- The tafsir of Ibn Jawzi (Zadul-Masir fi Ilmil-Tafsir), vol. 3, pg. 462, under the Verse of Purification.
8- The tafsir of Ibn Atiyyah (Al-Muharrarul-Wajiz), vol. 5, pg. 308, under the Verse of Purification.
9- The tafsir of Beydawi (Anwarul-Tanzil wa Asrarul-Tawil), vol. 5, pg. 12, under the Verse of Purification.
10- The tafsir of Muhyissunnah Abu Muhammad Al-Husein ibn Masoud Al-Baghawiyy (Maalimul-Tanzil), vol. 6, pg. 350, under the Verse of Purification.
Question 830 (website: 899) What is meant by the Ahlul-Bayt.
 Ahzab:33; Hud:73.
 Tabatabai, Translation of Al-Mizan (Farsi), vol. 16, pg. 465.
 In some hadiths by the infallible imams (as), they have referred to themselves as the Ahlul-Bayt. See: Tusi, Tahdhibul-Ahkam, vol. 5, pg. 232, the Jameul-Ahadith software.
 With help from: Translation of Al-Mizan, vol. 16, pg. 463.
 Ibn Kathir, Tafsirul-Quranil-Adhim, vol. 6, pg. 365.
 See: Tafsirul-Quranil-Adhim, vol. 6, pg. 365; Fakhr Razi, Mafatihul-Gheyb, vol. 25, pg. 168.