Warrior gets all the best weapons and armors, and has tons of HP. He doesn't have magic, but who cares. He's a one-man army. He makes the game easy.

Thief is much better in the remakes than in the NES version of FF1. He gets multiple attacks right from the start which makes him a pretty effective character. Thief has lots of agility which makes him good at running and dodging attacks. He still doesn't have magic though.

Monk is all offense. Don't equip a weapon on him, let him use his bare hands, and he has very high attack. His HP and defense aren't so great compared to the Warrior though, and he has no magic.

Red Mage is very average. He can equip lots of weapon and armor, but not as many as Warrior. He can use magic, but not as much as White or Black Mage.

White Mage is made for healing. Her attack and defense are terrible, except against undead where her white magic works well. She has high HP, and you can't over-emphasize how important healing is in any party, so don't rule her out.

Black Mage uses black magic to kill things. And he does it well. His physical attacks and defense and HP are horrible, but he can kill lots of enemies at once, which is fun to watch.

Knight uses even more armor and weapons than the Warrior, and also gets low-level white magic now, as though he wasn't strong enough.

Thief turns into Ninja, which is a major upgrade in many ways. Ninja has access to lots more weapons than Thief did, and also can use low-level black magic.

Master isn't a huge improvement on Monk; he doesn't get any magic, and can't equip much more equipment than Monk did. Master's strength is still his extremely powerful physical attacks.

(He also looks ridiculous.)

Red Wizard's access to weapons and armor keeps him relevant even though he never gets access to all the magic the specialist wizards do. He is still a very average character but he has his place.

If you want to use the highest levels of white magic, you need a White Wizard. Some of the upper-level spells are very nice. White Wizard still lacks in offense except against undead.

If you want to use the highest levels of black magic (and if only for amusement, you probably do), Black Wizard is your man. He still has horrible HP, physical attacks and defense, but he can explode things with magic which never gets old.