Desynthesis allows you to break down items into their material parts.
Fishermen/women can desynth (most) fish caught: This goes a long way towards the CUL desynth skills!
Crafters can level desynthesis skills that compliment their crafting speciality, e.g. if you like to craft carpentry items you will find CAR desynth skill the most rewarding (and easy!) to level.
That said, one can get a long way by purchasing items to desynth from the Market Board.
Desynthesis is not part of any crafting class: The desynth ability can be used no matter which class you are on, including DoW/M classes/jobs. But as with any class or job, desynth is something which you have to level. Even if you are level 60 CAR for example you will start out at level 0 CAR desynthesis skill. You can then level this up by breaking down items. It is best to work your way up as the difficulty of desynthing items needs to be at >50% chance of success to really make it worthwhile trying to desynth it.
A guide to desynthesis, and suggested items to use, can be found here.