Materia are magical orbs that you find throughout the game that can be implemented on a character to casts different kinds of magic. There are five classes of materia:

This is the most common materia used for everyday magic. Includes elemental, cure, and other forms of attack magic.

This materia is used mainly as a support materia. This kind is linked with other types of materia through linked slots on weapons and armor. Includes a materia called "all" used to make the linked magic materia affect all enemies if it is an attack magic or team members if it is a curative magic.

This materia brings up new commands or options on the battle field. These options are like regular attacks in that they don't require MP to cast. There is a special type of materia called Enemy Skill.

This materia is independent meaning it works own its own without having to be cast. These usually strengthen attributes you already have such as attacks, HP, MP, Speed, Luck, Exp, Gil, and others.

This materia is used to summon mythical creaturs to aid you in battle. Summon spells range from each end of the spectrum including attacks, magic attacks, curative magic, and status effects. Usually, summon spells can only be cast once per battle unless you have them leveled up. Every time you level this materia up, you can use the summon spell one extra time in battle.

Leveling Up
To level up materia, you have to use that particular materia orb often in battle and/or by gaining AP which many enemies give when defeated. The best way is to use a materia more often, and by leveling up, the spell becomes more powerful and effective. You can check its progress by going to the materia menu and seeing how many stars it has. There is a maximum of five stars per materia. Once it reaches five, you'll have mastered that materia and you'll get a replica of that materia as a bonus prize for your hard work. If you master all materia in a particular class (green, blue, purple, yellow, red), you'll have the option of gaining the master materia... a single materia capable of harvesting all the spells in that class. To get the master materia, you must have got the corresponding great materia color. Great materia are rare materia made by nature that you'll find as the game progresses. Later, you'll be able to store the great materia in the Cosmo Canyon observatory. To get the master materia, go up to one and it will react giving you your prize!

Enemy Skill
This particular materia belongs under the Yellow Materia class. However, instead of gaining experience, this materia allows you to "obtain" certain enemy attacks that you absorb during battle. Once obtained, you can now use the attacks as part of your inventory like a regular magic spell.