A simplified short definition of terms
First of all since the beginning of hadith sciences scholars made an evaluation of its authenticity and declared a hadith to be:
- sahih if they would regard it as authentic and correctly attributed to the prophet ().
- da'if if they had doubts.
- mawdo' if they are rather certain that the report is a fabrication.
and other terms were used.
The terminology modern scholars use was defined by ibn Salah or ibn a-Salah ابن الصلاح and even at his time scholars used different terms and defined them differently. And still some scholars would mix ahadith today in their works, therefore it is important to know whom made a qualification. For example in al-Silsiallah as-Sahiha of sheikh al-albani he would compile ahadith which are sahih or at least hassan according his own qualification.
However it is clear from the linguistic perspective (Arabic language) that the meaning of:
- Sahih: means a hadith has a high level auf authenticity (this doesn't exclude that there might be some slight weaknesses if we analyse the hadith or weaknesses claimed by some scholars due to the fact that they are less lenient in their qualification).
- Hassan: means it has a good level of authenticity, but there might be doubts for example on some variations of a hadith.
- Da'if: means it has a rather low level of authenticity.
The two sahihs
When Muslims (sunni-Muslims) refer to the two sahihs or the work of the two Sheikhs they mean the books which are attributed to imam al-Bukhari and imam Muslim and are commonly known as Sahih al-Bukhari and Sahih Muslim.
Their authors only chose ahadith which could fulfill a high level criteria each of the authors defined for himself to be a measure for regarding a hadith as "sahih".
Regarding this scholar say that the condition of al-Bukhari are "harder" or "Higher" than those of Muslim. But they say that imam Muslim had a higher focus on the correct "Arabic" wording in his book. Having this said if you found a hadith which both scholars mentioned in their sahih book with the same narrator chain then the version in Sahih Muslim is more accurate then that of al-Bukhari.
There are other authors who also complied sahih hadith books such as ibn Khuzaimah and ibn Hebban, but their conditions for accepting a narration as sahih can be regarded as more lenient than that of al-Bukhari and Muslim.
Therefore the two sahihs are regarded as the most authentic hadith compilations and sahih al-Bukhari is said to be the most authentic book after the qur'an as stated by many scholars. Long before the compilation of this book imam a-Shafi'i claimed that al-Muwatta' of imam Malik is the most authentic book after the qur'an.
Anything which is referred to by Darussalam is a reference to an Edition of a printed (and translated) copy of a book. And as stated on sunnah.com:
We are working hard to add grade information for each hadith not in the Sahihain (Sahih al-Bukhari and Sahih Muslim). At this point we are displaying grade decisions by Shaykh al-Albani and Darussalam (Hafiz Zubair 'Ali Za'i). Eventually we hope to have grade assignments from several other distinguished muhaddiths such as Shaykhs al-Arna'ut, Ahmad Shakir, and Abu Ghuddah wherever applicable.
So any qualification in the Darussalam prints is a qualification made by the scholar Hafiz Zubair 'Ali Za'i. This qualification can contradict that of known scholars who have qualified the Musnad Ahmad such like Sho'aib al-Arnaout شعيب بن محرم الألباني الأرناؤوطي.