I'm putting IV and VI together too, since alot of the reasons are the same. First of all, the storylines are great. It isn't often that an RPG storyline can make me give a crap about characters, let alone one from the 16-bit era. But there were a few. IV and VI were really good in that regard. The characters were likeable, the plots were pretty deep for their time, and the villain of FFIV was pretty interesting too, being related to the main character (yes, it's the Darth Vader Complex, but it works in this case).
FFVII really needs no introduction, since it's on most peoples' lists. And it should be. It's a great game. Granted, if you played it nowadays, the graphics and visuals are extremely dated (the characters look like dead puppets in most of the FMV sequences). But when this game came out, it changed everything. Literally, everything. And it's a good and bad thing. It's a good thing, because it pushed console RPGs, which at the time were niche, into the mainstream. No longer did they play second fiddle to action games or platformers. Now they were a big deal. Plus, change is important in an evolving industry. It's what keeps things fresh and interesting. However, it can be a bad thing, because with this change, we've left the old school era of Final Fantasy, and really RPGs in large. Once you change things that much, there's really no turning back.
FFIX was great simply because it felt very retro. Not only did it pay homage to the classic games, but the difficulty of the game required you to level grind and do all the old school stuff you used to do in the 8 and 16-bit era. Plus, the storyline was pretty interesting too. Even if half the towns got destroyed minutes after Zidane got to them...