For item drop mechanics, read this post:
https://www.tapatalk.com/groups/diablothehell/viewtopic.php?f=1&t=1683&p=19104&hilit=brand#p19104 - credits to Bue/Brad

As for unique item drops: bosses are 4 levels higher than the regular monsters they are based upon. So a horror boss on a 59 mlvl base (i.e. his mlvl is 63) can drop a unique up to qlvl 63, which is the highest qlvl (given that he can drop the required base item of course). The reason is because the quality of unique items is solely determined by monster level: qlvl <= mlvl. Also, it is worth to note that you always get the highest possible qlvl unique item dropped.

Base item drop restrictions for Purgatory difficulty (credits to Duke). They don't drop if their quality level (qlvl) is lower than the given values down below:
Helm:        qlvl < 8
Light armor:    qlvl < 8
Belt:        qlvl < 7
Boots:        qlvl < 9
Gloves:        qlvl < 9
Shield:        qlvl < 6
Axe:        qlvl < 7
Sword:        qlvl < 7
Mace:        qlvl < 12
Claw:        qlvl < 11
Knife:        qlvl < 11
Mallet:        qlvl < 11
Trap:        qlvl < 12
Flask:        qlvl < 12
Bow:        qlvl < 11
Staff:        qlvl < 7

Item drops restrictions for Doom difficulty:
Helm:        qlvl < 12
Light armor:    none
Belt:        qlvl < 26
Boots:        qlvl < 23
Gloves:        qlvl < 28
Shield:        qlvl < 12
Axe:        qlvl < 25
Sword:        qlvl < 20
Mace:        qlvl < 30
Claw:        qlvl < 25
Knife:        qlvl < 25
Mallet:        qlvl < 25
Trap:        qlvl < 24
Flask:        qlvl < 24
Bow:        qlvl < 20
Staff:        qlvl < 19

Items not listed (such as jewelry or medium/heavy armor) are not bound to such restrictions, any qlvl of them can drop in any difficulty if the base stats of the player allows it to. Except for some specific exceptions. In Horror difficulty, monsters' level (mlvl) needs to be equal to or greater than the qlvl of the base item for it to drop. In Purgatory, qlvl receives a lifting of 15, so mlvl 10 monsters can drop up to qlvl 25 base items there. Doom has no qlvl restrictions based on mlvl for base items.

- credits to Duke