Hadith Scholars differed over what constitutes weakness in Hadith and although they generally agree on the majority of the criteria, ultimately it is a matter of Ijtihad.
Al-Albani's Criteria for grading the Hadith were compiled in a book entitled Minhaj Al-Albani Fi Takhrij in 480 pages by Shaykh Muhammad Ahmad Uways Abdil-Karim which he submitted for his doctoral thesis. Abu Tahir Muhammad bin Zubayr bin Mujaddid Khan (from the Pashtun tribe of Ali Za'i) briefly mentioned his criteria in his introduction to Darusalam's Sunan Ibn Majah as well as his Tahqiq of Mishkat Al-Masabih. To mention these criteria here would be quite lengthy and unless you are proficient in this science, you must accept their (or at least someone else's) gradings. Allah said, "Ask the People of Knowledge (lit. who remember) if you don't know."
The two scholars rarely disagree on the grading of a narration and the majority of the time the difference is in the way they chose to classify it. Most of the time Ali Za'i grades the Isnad (only) when a Hadith is not present in Bukhari or Muslim while Al-Albani gives the final grading. An example of this is when a Hadith has a weak Isnad and Ali Za'i classifies it as such but Al-Albani knows of a support for it and raises it to being Hasan Li Ghayri (Weak but Hasan due to support) in which case he simply says "Hasan" while Ali Za'i says "Isnad Da'if"'. Darusalam didn't distinguish between Hadith Da'if and Isnad Da'if at the end of each narration because they felt the average Muslim is not in need of this distinguishment.
Ali Za'i specifically had a criteria which caused him to differ with Al-Albani and it is when a narrator was unknown and two Imams authenticated him, Ali Za'i classified him as Hasan Al-Hadith while Al-Albani did not take this stance. It is the cause for the difference in the grading over a report which occurs in Nasa'i and Abu Dawud in which Malik bin Numayr Al-Khuza'i reported that Allah's Messenger (peace be upon him) bent his finger when pointing. Another distinguished criteria of Ali Za'i that Al-Albani disagreed with was his general weakening of the Mudallis (the one who masks the identity of narrators in the chain).
One should not repeat everything they hear especially when it concerns scholars and Hadith Scholars most importantly. Al-Albani engrossed himself in research and substantiated his claims and mentioned the earlier scholars who agreed with him in his 34 volume Silsilah Al-Hadith As-Sahihah wa Da'ifah wa Al-Mawdhu'ah. The names of his teachers and subjects he studied under them are available in his biography.