The one that started it all.
Instead of the oft tedius and confusing controls of previous RPGs, the control scheme was very straightforward; instead of an ambigious plot and not really knowing where to go or what to do, there was a clear storyline with well defined characters you met along the way. The combat system, while having it's share of problems (an insane hit/miss ratio early on, losing your turn if an enemy died before you attacked, etc), was a nice change from what previous games in the genre had offered.
One of the most ingenious marketting schemes of the old NES era I can think of involved the dedication of an entire edition of Nintendo Power to Final Fantasy. Instead of getting the usual issue, suscribers received a strategy guide for Final Fantasy. It was a great guide, even if it arrived almost a month late, and many of us had already finished it... Ahem. But yes, it got the word out to the masses and shortly after Final Fantasy was a household name.
Of course, a strategy guide alone does not hook someone onto a game. The game itself has to be good, and Final Fantasy was. It provided countless hours to gamers, young and old and terrific replayability, being able to choose from the different classes and then seeing their upgraded form later in the game. Being able to say you beat the game with 4 white mages was the ultimate way to show off amongst other gamers. The edition of the mini-game (where you pressed a&b 50 times while on a ship, if I remember right) was a cute touch.