You can equip materia to your party members by opening up the menu and selecting "Materia". Materia can be equipped to weapons or armor, as long as there are free slots available.
Weapons and armor can have up to 8 materia slots each, allowing you to equip up to 16 materia per party member.
There are 5 different types of materia: Magic, Command, Summon, Support, and Independent. Magic materia lets you use magic in battle, and is green. Command materia lets you use different commands in battle, and is yellow. Summon materia is red, and allows you to summon powerful allies to help you in battle. Support materia is blue - this kind of materia works with Magic, Command or Summon materia to enhance what you are able to do with Magic, Command or Summon materia; this is called a materia "combination". In order for this to work, you need to have a pair of "linked" materia slots. Weapons and armor can have either single materia slots, which look like a circle, or linked materia slots, which look like 2 circles connected to each other. To create a materia combination, place either a Magic, Command, or Summon materia in one of the linked materia slots, and a Support materia in the other linked slot.
Cloud's Buster Sword has 2 linked materia slots. Some basic combinations to put in these 2 slots are, for example, Lightning-All or Restore-All. All is a Support materia, while Lightning and Restore are Magic materia. Lightning-All will allow you to target multiple enemies with a Bolt spell. Restore-All will allow you to use Cure on yourself and your allies.
Note that different Support materia aren't guaranteed to work with every single Magic, Command or Summon materia. For example, "All" materia (mentioned above) doesn't work with Command materia.
Independent materia is pink/purple. Independent materia grants passive improvements to your party members, such as raising their HP by a certain percentage.
N.B. It should be noted that there is a tendency for both Summon spells as well as Enemy Skills to be overpowered when compared to similar "standard" Magic spells.
Best Materia Combinations & Chains
It's very important to obtain all 4 Huge Materia, as you play through Disc 2, if you want to really max out your materia setup. After acquiring them, you can "visit" the Huge Materia in Cosmo Canyon using the planetarium machine.
If you have the patience to master the required Command materia and exchange them for Master Commands in Cosmo Canyon at the yellow Huge Materia, you can come up with some powerful combinations. Double Cut and Slash-All materia won't have any extra effect if linked to Support materia. But there's a trick – if you have Master Command materia linked to Support materia, the effect of the Support materia will carry over to your default attack, including onto 2x-Cut, 4x-Cut or Slash-All. You can have 4 Master Commands in a chain on one character, one each linked to HP Absorb, MP Absorb, Added Cut and Steal as Well. If you also had a mastered Double Cut materia equipped, it would mean if you use 4x-Cut, you would absorb HP and MP (because of the HP Absorb and MP Absorb materia), as well as stealing (because of the Steal As Well materia) and having an extra attack (because of the Added Cut materia).
Another combination is to put Hades-Added Effect into your weapon and also equip a Mega All materia. Even if your level is still quite low, getting through weak to medium enemies will be a breeze as these enemies are likely to be affected by at least some of the status attacks which Hades inflicts.
There's a flip side to Hades-Added Effect. If you equip Hades-Added Effect, or Contain-Added Effect, or all 4 of these to your armor, you can get such good status immunity that the party member won't need a Ribbon equipped; allowing you to use an alternative powerful accessory.
Here is a particularly powerful one. Match up multiple mastered Counter Attack materia on one of your party members and also equip a mastered Cover. You'll counter multiple times every time you get hit, and because you have a mastered Cover equipped, you'll be hit frequently. Cover plus 15 mastered Counter Attacks is usually overkill…
This is maybe my favorite. Equip a mastered Comet materia linked to a mastered MP Turbo materia and a 2nd mastered Comet materia to a mastered Quadra Magic materia. Then equip a W-Magic materia. Comet 2 consists of 4 continuous hits so with Quadra Magic linked you're looking at a chain of 16, then with W-Magic a chain of 32, while the linked MP Turbo raises the damage of each hit.
Let's look at some really powerful combinations. How about a Master Magic linked to an All, a 2nd Master Magic linked to a Quadra Magic, a 3rd Master Magic linked to an MP Turbo, and a 4th Master Magic linked to an Added Cut. Then also equip a W-Magic materia and an MP Plus materia; all on one party member.
Or try this: a Knights of the Round linked to an HP Absorb, a 2nd Knights of the Round linked to an MP Absorb, a 3rd Knights of the Round linked to a Steal As Well, and a 4th Knights of the Round linked to an MP Turbo; and then throw in a W-Summon.
Last but not least, there is the classic Revive-Final Attack or Phoenix-Final Attack pairing. These allow you to auto-revive upon K.O.
Levelling Up Materia
There are 2 really good places to level up materia. The better area is only visitable towards the end of FF7; but if you're trying to make your party really powerful, then the time to do this is near end-game.
Nonetheless, in Disc 2, Mideel Area on the World Map is a good option. You can actually reach some of the areas near Mideel when you're still in Disc 1 once you have the Tiny Bronco vehicle; here the "Spiral" enemies will provide a decent number of AP (80 AP each; so you can earn 240 AP for finishing off a group of 3 Spirals).
But in order to really max out a materia setup, you will want to go Inside the Planet in Part 3 of the game. Follow the left path where your party splits, and then take the up path. Here you encounter Magic Pots; Magic Pots will always demand Elixirs, so be prepared. You will want to use the "W-Item Cheat". Each Magic Pot you defeat will earn you 1000 AP. In the same area of the Northern Cave, you can encounter Movers. They always appear in a group of 3, and defeating all 3 will earn you 2400 AP.
Here's where it starts getting tricky. If you're good at math, read on. Otherwise, skip to the list below. Equip Cloud with the triple growth, 3 materia slot Apocalypse sword. 3 (triple growth) multiplied by 3 (number of materia slots) gives an efficiency of 9. Cid's Scimitar is interesting, as it also has triple growth, but in this case 3 (triple growth) multiplied by 2 (number of materia slots on the Scimitar) only gives you an efficiency of 6. Let's consider 1 (normal growth) multiplied by 8 (number of materia slots) - e.g. the Wizard Bracelet; or 2 (double growth) multiplied by 4 (number of materia slots) - e.g. the Rising Sun. Either of these 2 gives an efficiency of 8. An excellent option is Cid’s 5 materia slots double growth weapon, the Javelin, which gives the highest efficiency: 2 (double growth) multiplied by 5 (number of materia slots); 10. But let's say there were only 5 different materia you were interested in levelling up - in that case it would make most sense to equip the Apocalypse and the Scimitar, and get triple growth on each of those materia.
Want to know the theoretical most powerful materia combination? All 3 of your party members must be able to equip 16 linked materia each – equip weapons and Wizard Bracelet armor accordingly. Equip a Master Summon and a W-Summon to Cloud. Then fill out the rest of Cloud's slots with Mime-Counter. The Counters should all be mastered. Then give your other 2 party members 8 Mime-Counter each, and make sure all 3 party members have their Magic stats at 255. The idea is this: Cloud uses W-Summon with Knights of the Round. Then if immediately afterwards the enemy attacks all 3 of your characters at once, you would counter with a huge, 23-long chain of W-Summoning Knights of the Round.
Providing everyone's Magic stat is at 255 (here's how, and be aware it is not easy), you would be taking off either 9999 or very close to it with each hit (You can guarantee taking off 9999 with every hit if you alternatively link the Master Summon to MP Turbo. This raises the MP used by KOTR to 255, and any magical attack using 255 MP is guaranteed to hit for 9999. But it means you lose one Mime-Counter slot, so I don't believe it's worth it).
You should deal just short of 6 million HP damage (9999 HP x 13 Knights of the Round Table x 2 W-Summon x 23 Mimes). I've obtained all of the other combinations in this guide, but not this one.