Equipping materia is generally fairly simple. Naturally, you must first find the materia, although sometimes materia are automatically included in a character's initial equipment. Go into the menu and select Materia. You will see each character equipped with a weapon and armor, each of which generally contains one or more slots for equipping materia. The minimum number is zero and the maximum is eight. To equip a Materia, simply select a slot. If there is already a materia in that slot, it will be removed and moved to the menu. If you already had a materia selected in the menu, then that materia will replace the one originally in the slot. If not, select a materia from the menu to place in the slot. The menu also offers the option to remove all materia from a character at once. Note that equipping or removing a materia will usually alter a character's statistics. Be careful not to equip so too many materia that dampen statistics, particularly HP, such that a character becomes too weak. However, once you have visited Cosmo Canyon, the costs of such materia can generally be offset by equipping one or more HP Plus materia.
At the end of most battles, characters receive both AP and EXP. The amounts for each of these displayed are the total amount that each character in the battle party receives. Characters not present in battle, also receive a smaller percentage of the EXP gained, however, this is not applicable to AP. EXP allows characters to level up, while AP applies to materia. The AP earned at the end of each battle is applied to each materia that every character present in the battle has equipped. It is not spread out over all of the materia, but rather the amount displayed at the end of the battle is applied to each materia equipped. Each materia has a certain number of "star levels" and once a certain amount of AP is obtained, that particular materia becomes stronger, but in general it will also take more AP to reach the next star level. Additionally, all normal materia, with the exception of Underwater can be "mastered." When a materia is "mastered," that means it is no longer capable of gaining any additional AP. Also, once a materia is mastered, a brand-new materia is born that has zero AP. Materia that are mastered generally either sell for large amounts of gil, or sell for 1 gil, generally in the case of materia such as Summon materia, some Command materia and most Support and Independent Materia. On a final note regarding this subject, sometimes materia increases in strength when it is mastered (usually in the case of materia that increases statistics) while in other cases it simply maxes out the materia and causes a new one to be born. Finally, materia sometimes grow differently depending on the weapons or armor they are equipped to. Materia growth levels include Nothing, Normal, Double and Triple.
Nothing: Found on all characters' Ultimate Weapons with the exception of Aeris' Princess Guard. Also found on certain weapons which contain no materia slots. Materia equipped to these will not earn any of the AP displayed at the end of a battle. Recommended only if you feel you've already developed your materia enough, have armor equipped sufficient to develop the materia you need to, or if you have materia such as the various Master materia which can not gain AP.
Normal: Found on most weapons and armor and all characters' initial equipment. What you see is what you get with this. If it says at the end of the battle that you've gained 15 AP, then that is what will be applied to all materia equipped on weapons and armor with this growth rate.
Double: Found occasionally on weapons and armor throughout the game. The best weapon with this growth rate is Yuffie's Rising Sun. Materia equipped on a weapon or armor with this growth rate will earn twice as much AP as that displayed at the end of a battle.
Triple: on two weapons only: Cid's Scimitar and Cloud's Apocalypse. Cid's Scimitar is easy to obtain. It's in a treasure chest in an area near the submarine after you fight the Carry Armor during the Huge Materia mission in Junon. Apocalypse is slightly more difficult. Complete the quest in the Ancient Forest near Cosmo Canyon and you will find it in a treasure chest at the end of it. Materia equipped on either of these weapons will earn three times as much AP as that displayed at the end of a battle. Cloud's Apocalypse has two materia slots linked together, whereas Cid's has three non-linked materia slots.
Linked Slots and Support Materia
Support materia have no effect in and of themselves. In order to work, they must be paired with another materia, usually Magic, but sometimes Command and occasionally Summon. Throughout the game, look for weapons that have two materia slots that look like they're "linked" together in order to use Support materia. All characters' Ultimate Weapons with the exception of Aeris' Princess Guard contain four sets of two linked slots (although they also have no materia growth) and the armor Wizard Bracelet also has this. Support materia provide a number of beneficial effects, see the section on them for more details.
Magic materia are a green color and there are a total of 21 in the game. Equipping any Magic materia generally immediately provides a character with access to a certain magic spell, with the exception of certain very powerful ones, which must first earn AP before a spell becomes available. In general, earning more AP on such Materia will generally result in earning a more powerful of version of the previous spell.
Spells fall into three categories: Attack, Curative, Indirect.
Command materia are yellow in color and there are a total of 13 in the game. These materia add commands directly to a characters' menu in battle and therefore can provide a number of options that wouldn't otherwise be available. These range from the ability to sense the statistics of a target in battle to being able to directly copy a previous characters' actions. In most cases, earning more AP on these materia will simply serve to master them, however it may open up additional commands. One of them, Enemy Skill, actually does not earn AP, but rather provides a large variety of useful abilities and spells that are so useful they are worth their own subsection.
Summon materia are red in color and there are a total of 16 available in the game. They provide access to a variety of powerful summoned monsters, something which have been a tradition in Final Fantasy games since the fourth game. Unlike many FF games, where the summoned monsters are an essential part of the plot, the characters don't really talk about them in Final Fantasy VII, instead, focusing on the effects of using materia overall. Find any summon materia and equip it to a character and that character is immediately given the option to use that summon once in any battle. Then, earn AP on these materia to have a character be able to use the summon up to five times in one battle. Note that if you equip the same summon materia to multiple characters, then each character will be able to use the summon the number of times that their particular materia has been leveled up to. Note that many summons have elemental effects, so check the element of the monsters you are fighting before summoning, so you don't heal them. Some summons also cause powerful status effects. Unlike magic spells, summon spells can not be turned on your own characters, they can only be cast on enemies.
Support materia are blue in color and there are a total of 13 available in the game. They are special in that in and of themselves they have absolutely no use. Nevertheless, if used properly they can be some of the most powerful materia in the game. Support materia required a linked slot on a weapon or armor and one more materia to link them to. Early in the game, many players find it useful to link an All materia to one or more magic materia. All materia are also extremely useful in that they are easy to master and a mastered all materia sells for a whopping 1,400,000 gil. Later in the game, useful combinations include Final Attack + Phoenix (if a character is KOed, Phoenix is automatically summoned, bringing all KOed characters back to life) and Sneak Attack + Morph (best used in combination with a Mega All materia and Yuffie's Ultimate Weapon Conformer, makes the task of gathering status Source items much easier.)
Independent materia are purple in color and there are a total of 17 of them in the game. Independent materia don't provide any spells or commands, but instead usually provide some sort of status benefit, or occasionally a boost to the effects of other materia. They might also cause a character to do something they wouldn't normally, such as counter a monster's attack or take damage for an ally that's attacked.
Materia that doesn't fall into any of the above sections.