Fighter, Knight - Fi,Kn - This is the best attacker you can use. When he becomes a knight he can also use white magic up to level 3, and can equip all but 6 of the weapons in the game.
Thief, Ninja - Th,Ni - The thief is the most unnecessary class. It's like a fighter that can't fight quite as well. He can help you run away tho. When he becomes a ninja he can use black magic up to level 4 and well, he can attack better, how exciting. Hmm, I am being a little hard on him tho. He is good for running away, and he's fun to have. And the ninja is pretty cool. I must complain tho, that the Ninja's best weapon, the Katana, is a big (or should I say little) piece of crap, visually speaking. It's quite unimpressive looking.
Black Belt, Master - BB,Ma - The black belt is kinda crappy in the beginning. He becomes much, much better with leveling though. After a while you should unequip his weapon and eventually unequip his armor. He's the only class that can't ever use magic.
Red Mage, Red Wizard - RM,RW - The red mage is the über warrior of the group. He's a decent fighter and can use black and white magic. His problem is that he can't fight worth crap at the end of the game. He'll be hitting the big bosses for about 2 damage every other time. He's also fairly expensive, you'll be paying for weapons, armor, and spells. Oh yeah, no level 8 magic. :( I guess he's sort of a "Jack of all trades and master of none".
White Mage, White Wizard - WM,WW - The white mage is the protector and healer. In addition the level 8 spell FADE is the second most powerful attack spell in the game.
Black Mage, Black Wizard - BM,BW - This is the coolest looking warrior in my opinion. He's great for upping your attack power and killing all the enemies at once. Don't have a party without one.
You can win with any combination. Four white mages used to be considered the hardest. (I think it's actually four Thieves, if they go unpromoted.) Check out my White Mage Journal to see how I did it. With any party you choose unless the second character is a fighter he'll be dying a lot. I recommend that you have a fighter in your group. You should definitely have someone with white magic too. Still, you can pick anybody you want. Try four thieves or two black belts and 2 black mages or 1 thief 2 white mages and a black mage, whatever you want. Unless you want your ass trounced you ought to use one of these recommened parties the first time thru tho:
The conventionally recommended party is a Fighter, Black Belt, White Mage, and a Black Mage. The manual says it's "highly recommended". I spit on this party! Yes, it will get you through the game, but where's the fun? Go ahead and try it if you must, but I don't like it due to its recommendedness. :-)
Anyway, the party I like the most is: Fi, RM, WM, BM. It is one of the few parties that let you effectively use all the good weapons and armor at the end the way I like it. I recommend it for beginners. It has 2 people able to cast LIFE and EXIT which is very helpful for beginners. Also having 2 fighting characters, 2 white magic users, and 2 black magic users is very powerful. The flaws of this party are that it is expensive to equip and it's a little light on attack power later in the game.
Fi, Th, BB, RM: The default party on the NES. It has its charm, and is very powerful, but I don't like having only one magic user.
Another fun one is 2 fighters, a whitey, and a blacky. That'll kick ass thru the game. At the end one fighter has Xcalibur and one has Masmune. Ouch!!
Fighter, Thief, White Mage, and Black Mage is both the first party I ever tried and the first party I beat it with. It's also the default party on the Wonderswan. It's a good party but the Thief is a magnet for enemy attacks in the beginning.
The beauty of the game is that you can pick any combination you want. Early games, like this, the first Dragon Warrior, and the first Zelda, are a little unrefined and allow a lot more freedom than newer games.
For more party ideas, some savestates, and a running total of parties I've beat the game with go to The Party Page (non-groovy version).
You also pick a name at this point. Unfortunately, you are limited to only 4 letters per name (6 in FFO) so you have to be creative. I hacked a savestate to get more letters, you can see the results here and here. I'll eventually have a savestate editor that can perform this type of thing (among many others).
