Hadith commentaries are the equivalent of tafsir books when it comes to hadith sciences.
Basically these commentaries are called sharh شرح meaning explanation (plural shurooh شروح). And the content of these books can be somewhat different according to the intention and focus of the authors, for example some just try to explain words of a hadith, some check the narrator chain, some are extracting fiqh rules others emphasize on linguistic meanings, eloquence, some are quoting similar hadith from other sources, or differences in the wording of some versions of the same hadith or in the narrated version of the hadith collection itself etc..
Most major sunni hadith (sahih al-Bukhari, sahin Muslim, Muwatt'a Malik, sunan abi Dawod, sunan ibn Majah, sunan an-Nasa'i, Jami' at-Tirmdihi) collections have such commentaries, but there are also books that don't specify on a single collection for example al-Qadi 'Iyad's "mashariq al-Anwar" is a collection of rather ambiguous ahadith from the two sahihs and al-Muwatta'. There are also Mula 'Ali al-Qari's merqat al-Mafateeh which is a commentary on a hadith collection called meskhat al-Masabeeh ...
Note that later scholars often quoted views of former commentaries even if it was on an other book as there's a lot of intersection, for example you may find almost all ahadith of al-Muwatta' in the books quoted above.
Here's a list of some commentaries on specific books:
The most famous commentary is fath-al-Bari فتح الباري of ibn Hajar al-'Asqalani (the shafi'i scholar) Imam a-Shawkani was quoted with the statement:
لا هجرة بعد الفتح
There is no emigration after the conquest (one meaning of fath) ...
borrowing words of a sahih hadith.
But there's also an other fath-al-Bari of the hanbali scholar ibn Rajab which some specialist consider even better or showing more insight in the science of hadith.
Note al-Bari الباري is one of Allah's names or attributes meaning the creator. So both book titles may mean "What the Creator has given insight of" or maybe "What the Creator has unlocked", wikipedia says it is "vitory of the Creator".
'Omdat al-Qari عمدة القاري of al-'Ayni العيني the hanafi scholar.
Other well known scholars who have written commentaries and are often cited by ibn Hajar are ibn Battal ابن بطال a maliki mohadith born in Valencia (died 449 a.H.), al-Khattabi أبو سليمان الخطابي a shafi'i scholar (died 388 a.H.).
The most famous commentary is the one of Imam an-Nawawi the famous shafi'i scholar who also rearranged the Sahih of Imam Muslim into chapters and entitled them. He called his book al-Minhaj المنهاج (the guide).
Many Maliki scholars from al-Maghrib and al-Andalus were the basis of the commentary of whom step by step -hopefully- completed each others work as al-Maziri apparently died before completing his al-Mu'alim المعلم بفوائد كتاب صحيح مسلم, and al-Qadi 'Iyad died before completing his Ikmal al-M'ualim إكمال المعلم في شرح صحيح مسلم (Ikmal = completion), al-'Abi محمد بن خليفة الأبي later wrote Ikmal al-Ikmal based on the works of al-Mazri, 'Iyad and an-Nawawi before finally as-Sanusi محمد بن يوسف السنوسي wrote "Mukmil Ikmal al-Ikmal" (the finalization of the completion of the completion).
It is said that earlier scholars from al-Maghrib and al-Andalus considered Sahih Muslim to be more authentic than Sahih al-Bukhari.
Other scholars from other schools also wrote commentaries on Sahih Muslim like 'Ali al-Qari.
Many people think that we have only one narrated version of a hadith collection, when it comes to al-Muwatta' we clearly learn that this isn't the case. We have at least two versions which are nowadays well known the Version of Yahya ibn Yahya al-Laithi and the one of Muhammad ibn Hassan a-Sahybani (the student of abu Hanifa) beside others like the one of al-Q'anabi etc.. Note that the scholars of the other 6 books didn't rely on the first two copies at all, as both narrators were not accessible (Yayha ibn Yahya after the death of Imam Malik went back to al-Andalus) or are rather less known for hadith both Authors were beside Qadi's so they didn't have much time to teach and move to other places.
- Of course Maliki scholars are the majority on the list of commentators at first one must set ibn Abdal-barr ابن عبدالبر the famous muhadith of al-Andalus, he is the author of at least three commentaries at-Tamhid (>20 Volumes) about which ibn Hazm said the famous words:
وهو كتاب لا أعلم في الكلام على فقه الحديث مثله، فكيف أحسن منه؟
This is a unique book: I don't know any equal book in the science of hadith, so how would there be any better?
, al-Istidkar (which is more on comparativ fiqh based on al-Muwatta' discussing the view of the other fiqh schools primarily hanafi and shafi'i), beside a summary of at-Tamhid.
- There are also the books of al-Baji called al-Muntaqa, al-Kandahlwi's Awjāz-ul-Masālik ilá Muwattā' and Shah Wali Allah Dahlawi's "Al-Musaffa Sharh al-Muwatta".
- The most common commentary (4 Volumes) however is the commentary of Muhammad az-Zurqani محمد الزرقاني.
Some Commentaries on the sunan
Hashiyat حاشية السندي على سنن ابن ماجه as-Sindi نور الدين السندي (died 1138 a.H.) on sunan ibn Majah.
'Awn al-Ma'abod of Aabadi آبادي on Sunan abi Dawod. Al-Khattabi has also a commentary called ma'alim a sunan معالم السنن.
Sharah as-Suyuti on Sunan an-Nasa'i (he has commented on almost all the books quoted) which is called Zahr ar-ruba 'ala al-Mujtaba زهر الربى على المجتبى. The book of an-Nasa'i's full title is: "Al-Mujtaba mina as-Sunan al-Kubra" المجتبى من السنن الكبرى.
Tohfat al-Ahwadhi تحفة الأحوذي شرح جامع الترمذي of al-Mubarkfuri/al-Mubarakpuri المباركفورى (died 1934) on Sunan at-Tirmidhi. And ibn al-'Arabi has written "'Aridhat al-Ahwadhi" عارضة الأحوذي.