VI and VII are legitimately the best. I'm really surprised at how well VII holds up when played on PSP, actually; it's one of the few nostalgic games I've played that actually feels like it was better than I remembered it. Meanwhile IX is a good runner up, and X is kind of a mixed bag but still tolerable for the fact that it's got a really solid battle system. To summarize each...
VI has a steampunk setting where magic is dying and you play Terra, one of the last humans alive that can use magic naturally, as you fight with a group of rebels against an evil Empire that's hoarding the mystical power of Magicite and the spirits known as the Espers. The entire magic system is built around Espers, which you can call as summoned monsters. When you equip one, it brings several magic spells along with it that you learn as you level up.
VII is difficult to describe in terms of its setting's timeframe. On one hand technology in this world is somewhere around early 1900's levels. On the other, the discovery of Mako energy, siphoned from deep within the planet, gives the Shinra electric power company a HUGE boost and has allowed them to master much more advanced concepts like computers and helicopters. However, the only thing more powerful than Shinra's Mako technology, it seems, is Shinra's greed. Mako is the planet's life energy, and the more they siphon, the more they kill it off, and it's up to Cloud Strife and the eco-terrorist group AVALANCHE to make them see the truth... with explosives! The magic system of this game is based on Materia, IE special marbles you stick inside your weapons and armor. It's like the Esper system in a way, but much more flexible and compartmentalized.
IX follows Zidane Tribal and Tantalus, a band of thieves... ... well, actually they're performers... who have an important mission: kidnap Princess Garnet of Alexandria and bring her back to the kingdom of Lindblum in one piece. Things get more complicated, however, as they discover that Garnet's mother, Queen Brahne of Alexandria, is preparing to mobilize an army of dolls fashioned into the form of Black Mages. Where do these monstrous weapons of war come from? And how will this all end? You'll just have to play it to find out. This game's magic system is a bit less interesting than the previous two, but it works; you just get spells and abilities from pieces of equipment.
X is the story of Tidus, a Blitzball player from the futuristic city of Zanarkand, who finds himself transported 1000 years into the future. The city and life he knew is gone, and the world is terrorized by a colossal monster called "Sin," a being who supposedly exists to punish mankind for having the hubris to build machines; thus, Spira is regressed from the futuristic state Tidus knew and now comprised of a variety of islanders and tribes founded on the mysticism and dogma of the Yevon faith. Tidus travels with a summoner named Yuna on a pilgramage back to Zanarkand, hoping to find answers about what happened while Yuna seeks the power of the Final Aeon to defeat Sin. I make this sound cool, but the story's actually pretty weak, rather predictable, and full of odd holes. It does have one of the best turn-based battle systems ever devised, however, and has the distinction of Aeons, the game's summons, actually being full-blown characters that can take your place in a fight as opposed to just really big, flashy spells that cost too much.
