All enemies have a chance of dropping items. The exact odds of a drop from an enemy is 5%, the rarity of the item is then determined by 1-(0.11*(rarity-1))%.
Higher level enemies drop higher rarity items, though the exact milestones are unknown at this time.
Currently, there are two things that can boost item drop rate.
The Magnifying Glass (which can be bought for 15 Stolen goods from Air) which increases the drop rate by 25%, and the Lucky enchantment for acessories, which stacks (which will be explained further on) and increases the drop rate by 3%.
When in a party, only the highest drop rate is used (I.E. if someone has a glass, only their glass will be used, no matter how many other people use them). Then, Lucky enchantments are added from all party members (So if 3 people had both their accessories enchanted, the drop rate would then be 25% (Highest party member's drop rate) + 18% (Bonuses from 6 Lucky enchantments) for a grand total of +43%.
Some items only drop from specific enemies, such as 10* items and boss catalysts.