The conditions are those relating to the credibility level of the people in the narration chain.
First, a clarification: the conditions of Al-Bukhāri (Arabic: شرط البخاري) relate solely to his book Al-Jāmi' Al-Musnad As-Sahīh Al-Mukhtasar (Arabic: الجامع المسند الصحيح المختصر من أمور رسول الله صلى الله عليه وسلم وسننه وأيامه), commonly known as Sahīh al-Bukhāri. The same applies to the conditions of Muslim (Arabic: شرط مسلم), which relate solely to his book Al-Musnad As-Sahīh Al-Mukhtasar (Arabic: المسند الصحيح المختصر بنقل العدل عن العدل إلى رسول الله صلى الله عليه وسلم), commonly known as Sahīh Muslim (Arabic: صحيح مسلم).
Second, neither Al-Bukhāri nor Muslim at any time declared specific conditions other than what is already commonly accepted for authentication as defined by Ibn Hajar al-'Asqalani: continuous chain of narration by people who are trustworthy, able to memorize and preserve, and free of irregularities and detrimental flaws. Obviously, neither Al-Bukhāri nor Muslim would consider a severed or doubtful chain of narration for their sahīhs, so the conditions —as deciphered by scholars of hadith— always referred to the credibility of the people of the narration chain. You may refer to Fat'h al-Mughīth where As-Sakhāwī quoted An-Nawawi as saying neither declared their conditions (Arabic: لم يصرح أحد من الشيخين بشرطه في كتابه ولا في غيره). Several other scholars concur, but this is not a 100% agreed-upon conclusion as Al-Ghassāni, Qadi 'Iyād, and Al-Hākim had different views but I have not come across their proof.
Third, not all that is authentic is compiled in these two books. Therefore, the conditions of Al-Bukhāri and Muslim do not automatically translate into the authentication by Al-Bukhari or Muslim, i.e., neither authors nor any other scholar of hadith claim that all authentic hadiths are in their sahīh books. There are narrators that Al-Bukhāri considered authentic, but did not include them in his book for the purpose of brevity as he himself said: "I only compiled in my book Al-Jāmi' what is authenticated, and I left out from what is authentic so that I avoid a long [book]" (Arabic: ما أدخلت في كتابي الجامع إلا ما صح، وتركت من الصحاح لحال الطوال). Likewise, Muslim said: "Not every authentic [hadith] to me I compiled therein, but I compiled only what has been agreed upon [its authenticity]" (Arabic: ليس كل شيء عندي صحيح وضعته هاهنا إنما وضعت ما أجمعوا عليه).
Fourth, some hadiths had the entire chain of narration of credible persons, but were left out as the chain of narration may have been doubtful (either due to continuity of time or place. i.e., ارسال المكان أو ارسال الزمان). Both authors left out such hadiths, but others included them with the terms "trusted men" (Arabic: رجاله ثقات) or "men of sahīhs" (Arabic: رجاله رجال الصحيح). These hadiths are not automatically considered authentic.
Finally, to get a list of the narrators of Al-Bukhāri, the introduction of Hady as-Sārī Muqaddemat Fat'h al-Bārī, Ch. 6-10 (pp. 221-518), Ibn Hajar details all the narrators Al-Bukhāri used, partially used, or left out. This is a valuable read in Al-Jarh wa at-Ta'dīl, but way too long to include in an answer. As for Muslim, he used "agreed upon" to mean the agreement of Ahmad ibn Hanbal, Yahya ibn Ma'īn, 'Uthmān ibn Abi Shaiba, and Sa'īd ibn Mansūr al-Khorasāni. I do not know of a comprehensive list similar to that of Al-Bukhāri's, though.
Worth mentioning is that Al-Bukhāri was tougher in his selection of narrators for his Sahīh as he added a condition of a meeting of narrators, not just concurrently existing (Arabic: إشتراط اللقيا و ليس فقط المعاصره); whereas, Muslim was tougher than Al-Bukhāri in the preservation of wording (matn, Arabic: المتن). Therefore, it is quite typical to find that scholars consider Sahīh Al-Bukhāri to be more authentic than Sahīh Muslim, but when a hadith exists in both books, they tend to quote the wording of Sahīh Muslim.