If you've ever played Final Fantasy 7 the materia works much the same as there, except you can't remove it.
If you HAVEN'T played Final Fantasy 7, then materia is little stat boosting gems(basically) that you can insert into your gear. You have to have the correct crafter at the item's physical level to be able to meld materia, or go to a Materia Melder in town. You also need to have a Carbonized Matter equal to or superior in Grade to your Materia. IE, Peity I Materia is Grade 1, Peity II is Grade 2, etc.