Spiritbonding and Materia, weird terminology in FFXIV, but extremely useful to adventurers at the same time. To be honest, Spiritbonding is the act of creating Materia from gear and Materia is used to augment gear to give it additional states. Actually like gems in World of Warcraft.

When you earn experience by slaying enemies, crafting items or gathering materials, an item's Spiritbonding progress will increase until it reaches 100%. Once a piece of gear is Spiritbonded, you can convert it to a random Materia. However, converting a piece of gear that's appropriate for say a Pugilist or Rogue, would provide Material suited to those classes. 

Depending on the item level of the gear, it will provide different levels of Materia when converted. So a gear with item level of 60, can be converted to provide Materia IV. High quality gear will Spiritbond faster that normal quality ones, and for every Materia inserted, it will also Spiritbond faster; finding junk Materia off the market is perfect for taking advantage of this.

Any gear can be Spiritbonded, but only "convertible" gear can be actually consumed to produce Materia. Also, if you gain any Spiritbonding experience on an item, it becomes untradable to you; essentially soulbound.

Spiritbonding is available to all players, but Materia conversion can only be unlocked when a player gets a crafting class to level 19, and completes a quest located at The Bonfire in Central Thanalan.