The Final Fantasy series uses a different game play system with each game. The first few are related to the Dragon Quest series of games, but they move onto an active-time system later. The games don't have any connected story line and can be played in pretty much any order (save the sequels directly related to a line).
General consensus is that IV, VI, VII, and X are the best games of the series.
IV: Was the first FF game on the SNES, and considered by most Americans on where the series really gets going. This game has a number of firsts and unique features:
-The first FF games where characters were set in class and personality.
-On of the few FF games where you party, class and characters, are entire set by plot.
-With a max of five party members, IV has the largest active party limit in the series.
-The first FF game with the active time system, where you choice of action determines how long it takes to do. Also the first game where the enemies wouldn't wait for you to take your turn (though you could turn waiting on in a menu).
-The first where all spells where learned by either level or earning, no spells were bought in stores.
-The original US release was of the easy version of the game, the ports are all of hard type.
-The DS remake makes this the first FF with a remake as opposed to a port. The remake upped the difficulty above and beyond the hard-type ports.
-The earliest FF with a sequel.
VI: The last FF games on the SNES. Considered by some to be the best of the series.
-Has the most playable members out of any game in the main-line. (12 main, 2 hidden)
-Considered by many to have one of the darkest stories of the series.
-The first game where spells were learned through an addition to EXP called TP points.
-The first game to have limit breaks, that could be preformed at low health, but they were so rare that most players never see them.
-The original release had *numerous* bugs that reduced difficulty. The ports fix those bugs and are more difficult.
-The second half of the game was almost completely non-linear. Events could be completed in any order, or completely skipped.
VII: The first FF game on the PS1, considered by many to be the best in the series.
-You could only have three active members at a time, the first time that has happened in a FF game and a trend that continues to all future games.
-The first game where limit breaks were controllable.
-The first game where spells were non-permanent. They had to be equipped and if you removed them from your equipment you would no longer know the spell.
-The first FF to include pre-rendered full-motion video at key moments.
-The only FF with a prequel. Also the only one with an animated movie and OVAs.
-Also has a sequel which focus on one of the side characters from the first game.
X:
-The first FF game to allow in-combat character switching.
-The first to allow a character to theoretically learn all abilities and spells and use them all without class switching.
-The first FF to get a sequel and the first where the sequel is on the same counsel as the original game.
-The first FF not to include a world-map mode. The game remained in scale view the entire time.
-The first FF where summons remained in battle instead of as alternate versions of spells. A summon replaced the party until killed or dismissed by the player.
-The first FF since III to return to a turn-based combat system.
-Currently being remade into a HD version for the PS3. Whether this is a IV style remake or a simple HD port has not been announced.