This Final Fantasy raises the bar on story telling, with a fantastic battle system and soundtrack tied in.
The world (or worlds) that the game takes place in were the first big step for the company and for RPGs as a whole. The futuristic setting, along with ancient martial arts techniques, weapons, and customs, really mesh well together. This Final Fantasy provides the basic plot of any Final Fantasy after it (except for IX, in some respects), in that there's a group of rebels fighting against a higher power (usually the government or a military group, etc) in a futuristic setting while still using old weaponry. Plenty of themes started through this one which still hold true to the franchise as a whole.
The game starts off very well, introducing the main character and what's going on in the world, along with providing great scenery to get into the atmosphere of this world. I could go on and explain a lot of other details, but it's better just to play it for yourself and discover the world.
In my opinion, this game has the most creative and enjoyable cast of characters that any Final Fantasy has ever had (especially with a roster of 16 total, each having their own abilities). These characters all had depth to them too, which was very uncommon in RPGs before this. Each had their own back story that was revealed in some way, shape, or form as you progressed. In the second world, you don't even have to find all of the characters again, if you do not choose to, which is a plus. The game leaves a lot of adventures to be optional and a lot of characters are difficult to find (unless you happen upon them by mistake, used someone's knowledge, or are lucky). Same goes for a lot of the items and Espers (which continues to be my favorite system to date. Materia is a close second).
Overall, this game is just "magical." I didn't get to play this one until after Final Fantasy VII, Chrono Trigger, Super Mario RPG, and the like (since my brother got this one last. I started playing this when I was 10, and haven't COMPLETELY beaten it....but I'm super close. Got all the characters back and getting ready for Kefka soon). I mean, I had this game for the SNES (lost a cord for it), the PlayStation (wore a hole in the disc, so I can't play that one either), and am now looking for it again for the GBA. It just means that much to me to spend that much money and purchase a game that many times to be able to play it over and finally finish it.
In conclusion, if you're an "avid Final Fantasy fan" and you haven't played this game, then you really don't know what you're missing. And, for all you Final Fantasy VII fans out there, for a story more engaging and characters more unique and generally more awesome than those in VII, this game is a must.