Fighter
The Fighter is a fighting machine (as the name Fighter suggests). He has the most HP out of any class, and the ability to wear all the best armor lets him keep that HP for a long time. The Fighter also has access to most of the weapons in the game; his offensive power is matched by no other class (except maybe the BB). However all of the Fighter's strengths come from having the right equipment, and the best equipment is pretty expensive sometimes. The Fighter is the most expensive class to buy weapons and armor for, but that's more of a problem early in the game than later on. If you want an easy time in FF1, you pretty much have to take a Fighter.
Thief
The Thief lacks the strength of the Black Belt or Fighter, and he lacks the magic skills of the Mages. The Thief's main strength is his ability to run from battle and dodge attacks. In situations where running is the best option, the Thief is the best character to have, and there are MANY times in FF1 where you need to run (Sorcerers?). The Thief's also pretty inexpensive to equip; money saved on the Thief can be used on other players to help you through the game faster. Although he's not nearly0 as good a fighter as the Fighter or Black Belt, the Thief can hold his own in battle when he needs to. He's very weak at first, but at around Melmond you'll start finding better weapons he can use, and he keeps getting better from there. All in all, the Thief can be a good substitute for another fighting class.
Black Belt
The Black Belt is a pure fighter. He's the only character who can't use magic before or after class change, which is a minus in my book. Although he's not able to equip many weapons, and the weapons he can equip are very weak, the Black Belt's most powerful weapon is his fist. After level 7 or so, if you unequip the BB's weapon and let him use his bare hands, he'll start doing damage comparable to a Fighter. Because he doesn't require much (if any) equipment for most of the game, the Black Belt is cheap as dirt, and his extra armor and weapon slots can be used to store weapons and armor for other characters or magic-casting weapons like the Heal Staff. The Black Belt's weakness is his inability to use magic and his horrible defense, but if he has proper support from a White Mage, and you hide him in the back row of your party, the Black Belt might live long enough to be an effective fighter. (Sometimes.)
Red Mage
The Red Mage is the most well-rounded character in the game. He has access to the magic abilities of the White and Black Mages, and he can equip many weapons and armor not available to the Mages, making him strong in some regards as a Fighter. In the beginning of the game the Red Mage is going to be one of if not the strongest character you have. He can use the Silver Sword, and all the second and third strength elemental and healing spells, so he can pretty much do anything, at first. His usefulness starts to slacken as the game goes on though.
White Mage
The White Mage's main purpose is to heal her comrades. She lacks physical strength, being able only to equip Hammers, and will never be an effective fighter. Likewise, the inability to use armor other than Bracelets and some minor accessories make her very vulnerable to attack. In spite of her weaknesses, however, the White Mage is indispensible in her unique power to heal and protect her allies. Even the strongest fighter will die pretty quickly if you don't have enough white magic to keep him alive; the White Mage's job is to keep the fighters alive long enough to do their job, and this she does with more skill than any other class. Another strength of the White Mage is her strength against undead. White Mages are the only character that gets to use HARM magic, and there are a ton of undead monsters in the game, enough to make HARM magic worth getting. Although the White Mage's spells will be expensive in the beginning of the game, they're well worth the investment, and she makes up for it somewhat in that she doesn't need expensive weapons or armor.
Black Mage
The Black Mage is the weakest character, physically speaking. Black Mages have the lowest HP of any class and have pretty much no offensive power at all. However, his strength comes from his unique skill in wielding black magic. The Black Mage's use of black magic is matched by no other class; spells that damage all enemies at once, or kill an enemy instantly regardless of HP, are in some cases more helpful than even the strongest fighter. From a safe place in the back of the party, away from danger, the Black Mage can deal death to the enemy from a distance, quickly and efficiently.
Discord
Light Style