“Al-Sihah al-Sittah”, which is generally translated as "The Authentic Six". They were first formally grouped and defined by Ibn al-Qaisarani in the 11th century.
We Muslims view the six major ahadith collections as the most important. They are in order of authenticity:
1. Sahih al Bukhari, collected by Imam Bukhari (d. 870), includes 7275 ahadith
2. Sahih Muslim, collected by Muslim b. al-Hajjaj (d. 875), includes 9200 ahadith
3. Sunan Nasa’i, collected by an-Nasa'i (d. 915)
4. Sunan Abu Dawood, collected by Abu Dawood (d. 888)
5. Jami at-Tirmidhi, collected by al-Tirmidhi (d. 892)
6. Sunan ibn Majah, collected by Ibn Majah (d. 887)
And there are many other books of ahadees like Musnad Ahmad, Muwatta Malik, Sahih Al Jami, Mustadrak al Hakim, Ibn Hibban, Mishkat, Musand Abi Dawud Tayalisi, Muajjam at Tabrini kabeer, Muajjam e tabrani Ausad, Maujam e Tabrani Saghreer, Musnaf Abdur Razzaq, sunan Darmi, Darqutni, Shubul Eeman Baihaiqi, As Sunan Al-Kubra, Ibn Khuzaima, Targheeb wal Tahreeb etc
The first two, commonly referred to as the Two Sahihs(Sahiain) as an indication of their authenticity, contain approximately seven thousand ahadith altogether if repetitions are not counted, according to Ibn Hajar(May Allah have mercy on Him) 
Hadeeth or Hadees is synonymous with Sunnah, though the word ‘Hadeeth’ is generally used to refer to only the sayings of the Prophet Muhammad صلى الله عليه و سلم. The Hadeeth have been recorded by the Companions and collected in the books of Hadeeth. The word Ahadith or Ahadees is the plural of the word Hadith or Hadees.
Ahaadeeth are broadly classified into two categories
>> Sahih (Authentic)
>> Daeef (weak) 
More correct, opinion, is to call them “Al-Kutub Al-Sitta” (the Six Books [of Hadith]), instead of “Al-Sihah Al-Sitta” as was explained by a number of scholars, like Al-Hafiz Al-Suyuti in “Tadreeb Al-Raawi” (1/165), and in his explanation of the “Alfiya” (p, 17), as well as Al-Hafiz Ibn Katheer in “Al-Baa’ith Al-Hatheeth” (p. 33), and Al-‘Alamah Ibn Al-Wazeer in “Al-Rawd Al-Basim” (1/67), and Al-Imam Al-Laknawi in “Al-Ajwibah Al-Fadilah” (p. 66-140), and others”. For more, view scholars view given below.
Shaikh Ali Al-Halabi mentions in the Tahqeeq of “Al-Hitta fee Zikr Al-Sihah Al-Sitta” by al-Imam Sideeq Hasan Khan Al-Qanwagi (1248 – 1307 H) [May Allah have mercy on him], while commenting on the name of the book (p. 11), he said:
“As for calling the six books the “Sihah” (the Authentic [Six]) this is something that is to be criticized, since in it is a clear leniency that is not hidden. Al-Hafiz Al-‘Iraqi alluded to this in his “Alfiya” when he said:
وَمَنْ عَليها أطْلَقَ الصَّحِيْحَا * فَقَدْ أَتَى تَسَاهُلاً صَرِيْحَا
And who referred to them as the Authentic * Then he had come with a clear [erroneous] leniency
However the author i.e. al-Qanwagi [May Allah have mercy on him] quotes in another place in his book from al-Muhadith ‘Abdulhaq Al-Dahlawi [May Allah have mercy on him] that in his book “Lama’aat Al-Tanqeeh” (1/8-9), he gave an explanation to the use of the word “Al-Sehah” when referring to these six books, where he says: “and calling them Al-Sehah Al-Sitta is done as a generalization …”. I said: That is true, no doubt in it.
What is more correct, though, is to call them “Al-Kutub Al-Sitta” (the Six Books [of Hadith]), as was explained by a number of scholars, like Al-Hafiz Al-Suyuti in “Tadreeb Al-Raawi” (1/165), and in his explanation of the “Alfiya” (p, 17), as well as Al-Hafiz Ibn Katheer in “Al-Baa’ith Al-Hatheeth” (p. 33), and Al-‘Alamah Ibn Al-Wazeer in “Al-Rawd Al-Basim” (1/67), and Al-Imam Al-Laknawi in “Al-Ajwibah Al-Fadilah” (p. 66-140), and others”