I'll do a quick run-through of the characters. The order is somewhat by appearance, somewhat being as memory serves.

Terra: Terra is an exceptional fighter and magic user, and one of the two who can use magic early on. Later on she gains the ability to Morph, which can increase the power of her spells. She, too, can equip the ultimate weapons, and is an all-around good character to use.
Locke: Special ability is stealing from enemies, which can be useful. This can be upgraded to capture, where he steals and attacks. Locke is a fun character with quite a bit of a backstory. He is one of the four characters who can use the ultimate weapons, and is a pretty good fighter.
Edgar: Edgar is a good character early on because of his "Tools" command. He can use his tools to inflict heavy damage on either a group of enemies, or on one enemy at a time. Edgar can also equip the ultimate weapons, but I tend to be less inclined to give them to him as I prefer to use his tools instead and let someone else take the weapons.
Sabin: Sabin is an interesting character. He is a strong fighter, and his special ability is "Blitz", which allows you to unleash a special attack if you imput the right button combination. If you are skilled at using his Blitzes, Sabin is a good character to use, although his single-attack Blitzes target randomly which might make him seem somewhat less useful.
Celes: The forth character to be able to use the Ultimate weapons and the other character that can use magic at the beginning of the game, Celes, like Terra, has good statistics all around. Her special ability is Runic, which allows her to absorb magic attacks and gain MP based on the cost of the magic spell that was used, but she must have a sword equipped to use this ability.
Shadow: Shadow is a ninja, with good power behind his attacks and two rather interesting abilities. One ability is a completely random effect, his dog, Interceptor, will block an attack that targets Shadow, and then come out and attack an enemy for a good amount of damage. Shadow's actual ability is to throw items, there are several special "throwable" items that you can purchase, and just about any weapon you have can be thrown. Shadow can do a great deal of damage, so long as you can afford to purchase the materials for him to throw.
Cyan: Cyan is a Knight who can employ the Swordtech ability. A bar much akin to the ATB will begin to charge and depending on the number, Cyan will unleash a certain attack. As with Sabin, the target of the attack is completely randomized. Cyan has good physical strength behind him, but his attacks can be somewhat time-consuming.
Gau: Gau is an interesting character. He can "leap" onto certain enemies while in the Veldt. When he returns, he learns their abilities. In battle, Gau cannot employ normal attacks, he instead chooses a "Rage", in which he mimics the attacks of a certain enemy. Gau attacks randomly and automatically, when Raging you have no control over Gau.
Setzer: Setzer is a gambling man, and as such his special ability should come as no surprise. Setzer's "Slots" ability allows him to unleash several different attacks, depending on the result of the slot. Some of Setzer's attacks can be quite powerful, and some mimic the attacks of an esper. Setzer's slot has one fatal combination that is all but unheard of, but if you do happen to get it, you will lose immediately. It is an extremely rare occurence, however, and is not something to worry about. There is an item that allows Setzer to throw Gil instead of using his slots, and the damage dealt is the amount of Gil lost.
Mog: Mog is a moogle that has somehow learned to communicate with humans. His special ability is to Dance. Mog learns a dance by fighting within different parts of the map. When Mog begins to dance, he will act randomly and automatically, much like Gau while Raging. Mog's dances are all terrain-based, if you try to use a dance on a terrain other than the one that Mog is on there is a chance that he will stumble and fail to begin the dance. Dancing always works on the proper terrain. If Mog successfully uses a dance of a different terrain, the background terrain will change for that battle.
Strago: Strago is a crotchety old man who can master the attacks of monsters and use them with his "Lore" ability. If Strago sees an attack, he will learn it after the battle, though I believe he may need to be alive at the end to use it, someone else can correct me on this. Strago is good with Magic, and certain Lores, like the infamous Grand Train, are quite useful.
Relm: Relm is a young girl with the ability to cause her paintings to come to life. Relm can use her Sketch ability to sketch a monster of your choice. If the Sketch succeeds, Relm will mimic one of the monster's attacks. An item will allow Relm to take control of a monster instead of Sketch it. Relm tends to work well with Strago because she can sketch or control a monster and make it use an attack that Strago can learn. Relm's Sketch has been known to glitch, which makes some people disinclined to use her. Speaking from firsthand experience here, when I attempted to sketch the Red Dragon in the Phoenix Cave while it was invisible I suddenly received insane amounts of Dirks, 75 Illuminas, about 100 Paladin Shields, 8 Atma Weapons, and a large assortment of other useful items. This was on a cartridge game and attempting to duplicating it might not be the best of ideas, but be forwarned that the game can become messed up through the use of sketch.
Umaro: Umaro is a massive sasquatch that acts fully autonomously at all times. He is strong, he doesn't use magic, and there are two items that can add more attacks to his repetoire, but all of his attacks are random.
Gogo: Gogo is a rather interesting character. He has the ability to Mimic people, copying the attack that was performed before his. The interesting thing about him, however, is the fact that, by going into his status menu, you can give him any special ability that another of your characters possesses, which gives him a great deal of versatility. If you give him the magic ability, he will have the combined spells of the other three members of your party, but he cannot equip Espers which means he will not get the stat bonuses other members do.

Hope that helps!