Empty cocoons (or 'empties') are used for capturing minions in the forest. Mahbu will always refill your empties (up to the maximum of 12) whenever you access her Nagi magic. When you attempt to capture a minion in the forest, you lose an empty, regardless of if you succeed in the capture or not.
As stated by Garai "Empty Cocoons are normal silkworm cocoons. They have not been changed in any way."
Firefly cocoons are cocoons that contain a minion that has been captured in the forest. They are unusable as they are and must be purified in order to access the monster.
White cocoons are firefly cocoons that have been purified by a Nagi woman. During the game, Mahbu will purify your firefly cocoons and turn them into white cocoons so that you may use them. You can choose to use the minion as it was caught, or merge with other monsters. Or you can also spin the cocoons into fine silk that can be sold at the town shop.
A Black Cocoon is the result of a Nagi man performing the Ritual Of Purification instead of a Nagi woman. As a result, a great evil is born. The same ritual performed by a Nagi woman results in a White Cocoon, which a Cocoon Master can use to summon Minions.
Garai mentions that people will kidnap Nagi men and force them to perform purifications and then use the Black Cocoon minions.
Fairy cocoons are only mentioned in the 'Legend of Alcana' as told by Poto, the graveskeeper. In the story, Alcana spins fairy cocoons to turn into silk for her lover, Prince Menek. His father, the King, desires the fine silk and poaches fairy cocoons out of greed. He then boils the fairies alive within their cocoons in order to make the silk. It isn't clear how otherwise Alcana acquires the silk, but one assumption is that she has waited for the fairies to abandon their cocoons in order to turn them into silk. So it is maybe possible that these fairies have used the cocoons during a sort of pupal stage of their development.
It isn't really explained what exactly these fairies are.
- Due to the fact that he is unlikely to be Nagi, it is clear that The Poacher does not purify his own cocoons. It is likely that his minions are Black Cocoon minions. Due to Garai's mention of Nagi men being kidnapped and forced to perform purifications, it is possible that she is filling out this part of the story. Though The Poacher's background isn't clear at all.