Final Fantasy IV remains one of my favorite vidoe games ever made. The only games I would put in the same league are Star Wars: Knights of the Old Republic (Xbox, PC, or the Mac App Store), and the Portal series.
What made FFIV so engrossing is that it rounded up the best ideas put forth by its competition, reworked them into a new whole, and in doing so owned those concepts.
Final Fantasy IV shifted to a movie-like mode of storytelling, enriched the battle system through well-hidden complexities, has a plethora of locals and vehicles, and has some of the best music that I have heard in a video game. By using elegance to articulate its complexity, Final Fantasy IV manages to immerse the player in a compelling story, while at the same time making the gameplay experience itself more intense and memorable. This can best be seen in the presentation of the Active-Time Battle system:
Tokita says that the game's battle mechanics were inspired by F1 racing, the visual concept being that faster combatants would "lap" slower ones; this became more self-evident in later games, where characters' turn order was depicted through the charging of ATB meters that filled at a different rate for each character. But even in FFIV, where those details were hidden invisibly away, fights were nevertheless infused with the excitement of a race.
It should also be noted that one of its major strengths is that it jams an amazing amount of character personality into tiny two dimentional 16-bit sprites.
Final Fantasy IV is one of the few games that I have played over and over, as it has been released on multiple systems. I am hoping that it is released for the iPhone soon.
And do not forget the soundtrack (from iTunes) or the Sheet Music.