I'm a little late on the draw here, both because the game has been released for a while, and because the DS version is a re-release of a game that was in Japan in 1990. Still, it's what I've been playing, so I thought I'd share my thoughts on it.
This game, like every other Final Fantasy game, is an RPG (in the Japanese sense). That is, you're handed characters and guided through the story, but you get to run through the battles. Well, "story" is a bit generous. The first three Final Fantasy games were essentially about the battle system, finding cool spells and powerful equipment . . . basically like World of Warcraft, except with older graphics and no social interaction (if that's what you'd call what you get in WoW). The stories were just short segments of text inbetween quests to help you advance to the next part of the game. Even in that, it's pretty standard FF fair: Chosen heroes, find the crystals, stop the bad guy, save the world, blah blah blah.
So, the story's nothing to write home about, but the updated graphics look nice on the DS, and the battle system is pretty fun. In FFI, the only one released on the NES in America, you chose a character class for your party of four in the beginning of the game and that is how you played the game. In FFIII, you get to change your character class whenever you want. All stats will change to levels appropriate for that class when you switch, so there's very little downside to using whatever classes you want. This gives a lot of flexibility to the player and a lot of replayability to the game, since there are 23 different classes altogether.
The downsides to this system are the times when choices are distinctly narrowed for you, and the lack of equipment for some classes. Certain parts of the game put you in the position of choosing classes because those are the only ones that stand a chance of surviving the encounters. I suppose you could praise the game for forcing you to step outside your comfort zones, but I see it as taking away some of the choice you should have in playing the game.
Then there are times where a character might seem useful, but his equipment is so old, or in short supply, that he's worthless anyhow. An example is the Scholar. This class's most useful ability is that item effects are doubled. This makes attack items incredibly potent in a Scholar's hands. However, those attack items are few and far between, and do only elemental damage. As the game gets further along, most enemies will have no elemental weaknesses (and some elemental resistances), so the Scholar loses his advantages.
Still, that doesn't handicap the game too much. It's still fun, and there's still plenty of room to explore the game's variety. If you are a Final Fantasy fan, pick this one up. It'll keep you entertained.