Materia is the system used for characters to learn various abilities in the game. They can be equipped onto the various weapons and armors that characters wear, and they come in many different kinds (which are distinguishable by colors).
Some Materia have stats that can be granted to the wearer, which can either boost or drop certain stats. When you win battles, you will win AP, which is used to level up Materia. If a Materia levels up, the previous stats can be enhanced, the player might gain new abilities or can use the previous abilities more times. When the Materia reaches its max level, another Materia of the same kind is produced.