Fighter
The Fighter is true to his name, no more, no less. He is your typical powerhouse that can equip all kinds of weapons and armors. He has very high strength and vitality, and will eventually become very agile as you Level up, plus he gets tons and tons of Hit Points, seemingly to no end. The Fighter is good because he is a real survivor. He has high HP, and can equip lots of Armor and defense, so each Hit Point will be worth a lot. With the combination of high HP and high defense, the fighter can take huge beatings and still remain standing. However, he has very little speed at the beginning of the game, is rather dumb, and cannot use magic.
Fighter Knight
Well, the Knight is pretty much the same as the Fighter, save a few things. The Knight now gets to equip all kinds of Weapons and Armor. Now more than ever, he will be your star fighting character, and with these new weapons and armor, he is ready to pound anything, or take a pounding just as bad. What's more? He can use some White Magic!!! Well, it's not much, but having some extra cure magic does help. He can get magic such as CURE and CUR2, but his lack of intelligence makes his magic somewhat ineffective.

Thief
The Thief is, uh, not really a Thief. He can't really do much, and he doesn't have much Strength or Hit Points, but he is considered more of a fighting character only because he can't use magic. The Thief has a very high luck stat, but that only affects the ability to run (and even that I'm not completely sure of). So, while other characters are fighting and earning experience, the Thief is running away and staying weak. Since he never has a good attack, Hit Percent won't matter much. Evade won't do him much good, as most enemies will hit you no matter what. Also, with his low Absorb rating and low HP, any hit WILL be devastating. He does develop good speed later in the game, but not even as much as a Fighter. I strongly recommend against using a Thief, since he really has no strong points.
Thief Ninja
Oh my God!! The Thief actually became USEFUL!! The Ninja can equip almost every single weapon in the entire game, which significantly increases his firepower. Also, he can also handle lots of armor, giving him some good defense. The Ninja can also use some Black Magic, including FAST, probably the best magic in the game. However, other than this, he is little different than the Knight, except his strength is much lower. This, combined with how tough it was to actually keep him alive this long, cuts into his usefulness a whole lot.

Black Belt
The Black Belt is a very unique character. He fights very well with his bare hands, and gets good Hit Points, with OK strength, which will increase rapidly as you level up. The two most unique things about him are ABSORB and DAMAGE. For some reason, his DAMAGE always equals twice his level, provided he has no weapon equipped. Much unlike any other character as well, his ABSORB equals his level if he does not have any armor equipped. This is especially weird because most characters have an ABSORB rating of Zero without armor. While his Absorb can get up to 50, provided he is at Level 50, he does have pretty good defense. However, that doesn't really help much, because he will always suffer, as through the vast majority of the game he will NOT be at Level 50. What's worse? He can't equip much of any armor at all to compensate! All in all, his defense will always be fairly low. He really stinks at low levels, but the Black Belt can serve you well later in the game. The weirdest thing about the Black Belt is how he opens up. In the beginning, he doesn't get many Hit Points. But, since he gains Vitality every level, no questions asked, he will very steadily gain more HP every level he gets. Not only that, by level 8 or 9 or so, he gains the ability to very efficiently fight with his hands. While he will be weak against many powerful bosses or any other monster with high defense for a while, the "opening up" theme WILL kick in. While it will take time, by Level 30, he'll be able to do over 1000 damage to just about any monster there is! So, to make a long story short, if your Black Belt isn't good enough, level him up a bit, and he'll be a LOT better. He is also good because you will not have to spend much money on him, which can save lots of time. However, there is much Armor that he cannot equip, making him rather vulnerable to attack. He can do good damage without a weapon, but when you come to fight a boss with high defense, you will find him rather useless when he cannot do much damage at all unless he as at a VERY high level, and will always be very open to a boss's tough offense.
Black Belt Master
Well, this guy uh, almost doesn't change at all! He still can't equip many weapons or much armor, which really hurts. His low defense will always plague you, but his offense can be a different story. Technically, at Level 50 (the highest) it is "possible" to get 20 hits, with critical hit, causing over 8000 damage, so he can kill ANY guy in one shot. However, he can only do NEAR this much damage to the most wimpy and crippled monsters (even then he still never does 8000 damage), and he will still have a lot of trouble doing much damage at all to a boss or tough monster. He is also the only character in the game after the class change who can't use magic. I noticed when I was fighting Lich with Level 15 Black Belts that they hit 4 times, but only got around 30 damage! Even with a Critical Hit, they only got around 50. I'm not sure why this is, even though he can do tons of damage to just about any wimpy monster in the game. However, when I advanced him to a Master at around Level 30, he started doing around 1000 damage to any monster, regardless of defense! He really is a unique character, as always.

Red Mage
Now this is your star player. He has no super high stats, but likewise, no low stats. Hence, you will almost never find him in a situation that he cannot handle. The Red Mage gets exceptionally high Hit Points in the beginning of the game, although the rate of increase will slowly curve off, but not until much later in the game. Even still, he will always have enough HP to take some heavy beatings, although not quite like a fighter. Also, a Red Mage gets good Strength, especially for a Magic User. This Strength, combined with a good long list of weapons and armor that he can use, make him useful for fighting as well as using magic. Also, his agility, intelligence, vitality, and luck will always be above average. Now, besides being a pretty good fighter, the combinations of magic he can use are phenomenal. He gets both White and Black Magic, for the ultimate combination of attack, curing, and support magics, which can also boost the other characters' usefulness.
Red Mage Red Wizard
The Red Wizard, still your star player. Well, his usefulness drops off slightly toward the end of the game, when his HP gains start to slow, and he kind of becomes stuck in the middle. He is neither a super-power fighter, like the 3 fighters who now become very effective with the Fight command, and there is a lot of high-level magic that he can't use. However, he STILL is good. He will always be strong enough to knock and obnoxious enemy out cold, and the powerful combination of White and Black magic will always dominate.

White Mage
The White Mage is your typical healer in a classic RPG. As you would expect, he gets a lot of Intelligence and spell casting ability. He has very little offense, defense, Strength, or equipment that he can use, but he can use a plethora of support and healing spells that help both himself and other characters. He is good to have around on long trips when potions just aren't enough to keep your party going, or when you are facing a fierce enemy where you need quick defensive spells. He gets fair Vitality, and HP, but without being able to equip much good armor, he tends not to be very survivable, and you will often find is lack of offense frustrating in long caves and fortresses.
White Mage White Wizard
Well, he doesn't really change much. He still can't equip very good weapons or armor, and will always be somewhat weak. However, he can learn ANY White Magic spell in the game. There is one spell of the highest level that will actually give him some firepower. FADE will cause about 200-400 damage on all enemies, which can be helpful at times when you just want to get rid of a bunch of guys without worrying about defense, but rater how fast you can kill them all (4 Frost Dragons can never be good).

Black Mage
The Black Mage is also a typical dark magician in a good classic RPG. He has very high intelligence, and his powerful magic can cripple a big horde of attacking enemies in a flash. However, he only gets so many magic points, so you will have to conserve this on long adventures, and you will find that he has very little offense other than magic. Also, his HP, Vitality, and Defense are utterly pathetic, and you will often find yourself coughing up tons of cash for healing potions one after another, or exhausting other mages' magic to keep curing him. He really cannot equip much of anything at all, so you will find him about as survivable as a slug in a salt mine.

Black Mage Black Wizard
Much like the White Wizard, the Black Wizard does not change much through the class change. The Strategy Guide says he gains strength, but I really don't know where that came from. All stats remain the same through the class change, and he actually gains strength slower than he did as a Black Mage. The best thing is, he can learn ANY Black Magic spell in the game. Even still, his Strength, Vitality, and HP will always hold you back, and he always seems to be either dead or really close to it.