Materia extraction has been a thing since Final Fantasy XIV relaunched with A Realm Reborn, but a recent change has made it more beneficial than ever. As of patch 5.2 you no longer destroy a piece of equipment if you extract materia from it, and you can even pull the stat boosting orbs from high-end gear.
Whenever equipment reaches 100% spiritbond you can extract materia from it.
Once an item becomes entirely spiritbound to your character you can now pull materia out of it without destroying it. Just right-click/press Square/click X on the item to bring up the sub-command menu and select Extract Materia. You will be asked to confirm this choice, and after you extract the materia from the item its spiritbond will reset to 1%.
Do not confuse this with Retrieve Materia: that pulls materia you have slotted into your gear out, with a greater chance to fail the higher the grade. Extract Materia will be grayed out if the spiritbond is not at 100%, so you can’t accidentally reset the gear. From early testing it appears that materia grade is determined by the item level of the equipment you are pulling it from, with 450 gear giving grade seven materia and 460 gear creating grade eight. The type of materia is also selected by the type of equipment, so Machinist gear will give out aim materia, and Warrior armor will generate tenacity materia.
Keep an eye on your gear and extract materia whenever it reaches max spiritbond. Do so and you will find yourself saving gil on materia in Final Fantasy XIV. Seven know we burn through the little orbs without care.