Throughout your adventure, you will be asked to deliver letters from one moogle to another. The reason for this is because Mognet Central is down, so they can't deliver the letters themselves. It should be noted that you won't be able to do this side quest until Disc 4, and all of the letters prior to this have no bearing on the side quest itself. The side quest entails that you just deliver letters from one moogle to another, so it's nothing new, but you will have to do something extra in the end.
To begin the Mognet Central side quest, on Disc 4, head to Alexandria and visit the chapel. Inside should be three moogles, one of which is Stiltzkin, but you want Kupo. Speak to him and select the Mognet option, and he will ask you to deliver a letter to Atla. This begins the chain for the side quest, with the letters/locations being as follows:
- Kupo (Alexandria; in the chapel)
- Atla (Burmecia; in the far back)
- Mogryo (Black Mage Village; outside Chocobo’s stable)
- Kumool (Ipsen’s Castle; second room)
- Mois (Entrance to Fossil Roo; On Qu’s Marsh side and requires Quina in party)
- Noggy (Daguerreo; on second floor on right side)
- Kupo (Alexandria)
Upon delivering the final letter to Kupo, you are told that someone in Alexandria has the Superslick item to begin Mognet Central again. That's the only hint you're given, but if you head to Ruby inside of her theater, she will give you the Superslick. With that in your possession, you need to find Mognet Central, so you will need at least Choco with the Ocean Ability and a Dead Pepper. Head to the Outer Continent and from Madain Sari, go northeast to find an island. Look for the long mountain range on this island and find the U-shaped part of it. There will be a crack in the mountain there, so use a Dead Pepper while riding Choco to blast a hole in the mountain. Inside will be Mognet Central, so head to the second floor and speak with Artemicion, handing over the Superslick item. This will finish the side quest and you will be rewarded with a Protect Ring.