AHAHAHA! Nobody has mentioned Final Fantasy VII yet! I'm actually happy about that. (I'm guessing that is the "cliche" answer SwiftSign mentioned...) I just started playing it a month or two ago, and to be honest, I'm really not liking Cloud as much as I thought I would. All the stuff that happened with Sephiroth and Aeris and all was cool, but still...
Hm, I'm actually surprised that the popular favourite here seems to be VI. Although, I actually have to agree with that one. I just love all the characters in it (both the heroes AND the villains), and the gameplay feels very smooth and straightforward, though still a bit complex and always with new things to learn. And it's full of interesting and funny little moments--Kefka complaining angrily about sand (of all things), fighting with an army of Moogles, a very memorable opera scene about to be crashed by an octopus, a playable flashback of Terra's past, Setzer sweeping foes with a Chocobo stampede... It also has one of the catchiest main battle themes I ever heard. And hey, Setzer's airship is pretty darn cool too; it even has its own mini-casino inside (not that any of the games are playable).
I was never a fan of games where you have to choose a character from a certain class and choose what party to use at the very beginning, and I don't give a (insert nasty word here) about how fancy the graphics are in a game. And plus, I'm a little bit biased because I haven't played very many Final Fantasy games. I want to try out IV and the very first one next, but all I've played were VI and VII so far. I'll probably never play VIII onward anyway, simply because I don't have the right console to play them...