I know I'm probably going to get flamed for this, but in all honestly, "To each his own". I guess I should put a disclaimer for those to realize there is a definite divide in the final fantasy series in style and feel after FFVI. I'm definitely biased towards games that are of or feel more like a pre-FFVII style.

1.  Final Fantasy VI SNES(As FFIII in USA)/PS/GBA
   The final SNES FF is also the most epic in terms of story and leveling system. The magicite in this game is the granddaddy of Materia in FFVII.  
2.    Final Fantasy IV SNES(as FFII in USA)/PS/GBA/DS
   A classic title in the RPG series. There are a diverse set of characters, something that ends up becoming a staple in the series.
3. Final Fantasy IX PS
   I found it to be a great homage to the classic FF titles while incorporating the limit break system from FFVII with the equipment based skill learning system from Final Fantasy Tactics.  
4.    Final Fantasy V Super Famicon/PS/GBA(?)
   A job class system that is more complex and more fun than FFIII, but the plot is a step back from FFIV.
5.  Final Fantasy X PS2
   A candidate for title "only FF game the masses have played". Like many RPGs of the PS2 era, this game is pretty linear and is chock full of cut-scenes a la the MGS series. Despite this, the game has a unique grid based leveling system and a nice modification to the limit breaks from FFVII.  
6.  Final Fantasy 3 Famicon/DS
   A fun little RPG that lets you switch jobs on the fly. The endgame made me want to off myself, although it is far from the hardest endgame in the series.
7.      Final Fantasy XII PS2
   Take FFXI, take out the job system, add in a leveling system that resembles FFX, add in the summons from FFVIII, and make it single-player and you have FFXII. Since I'm not ranking FFXI it should be known that I loved the difficulty of the game. Not having to rely on my fellow americans to keep off aggro and heal made the post-endgame enjoyable.
8. Final Fantasy X-2 PS2
   A needless entry to the series and the first direct sequel. Despite playing for 30 hours just to get to the answer of one question you only found on youtube to begin with (because Square just loves us americans), this cannot be ignored as being the FF title with the fastest pace in battle. Switching classes in this game is so fun.    
9.  Final Fantasy VIII PS/PC
   Yes, this game was stupid in that you can idle in a place where time stopped and still earn income, but besides the odd/predictable plot, it was a good game. This game introduced summons having their own HP and drawing magic from enemies added a nice twist to the mix (mashing square? not so much). And FYI, play the game before you decides who actually said the text in the game's opening    >_>
10.  Final Fantasy 1 NES/PS/GBA/PSP
    A generic, but very difficult RPG with a D&D magic system to confuse the hell out of anyone who hasn't rolled the dice. I think the main reason for the low ranking because there are many other games with a better complete package.    If you fail at life at this game, try the GBA port (not sure the psp has the following) it removed the D&D spell system with the MP that we so know and take for granted.
11.  Final Fantasy VII PS/PC
    Think a sci-fi/emo FFVI with graphics that don't age well. This was a turning point of the series with the art direction going from medieval/fantasy to sci fi/fantasy. I'm probably one of the few that hated it, despite it bringing the limit break system.    
12.  Final Fantasy 2 Famicon/PS/GBA/PSP
    The absolute worst rpg-leveling system ever created. I played through this more to be a completist and less for the entertainment of the game.  
13.    Final Fantasy XIII PS3/360 
    Remember when I called FFX linear? I take that back, this game takes the cake. "The game becomes more open later on..." blah blah blah. Besides the linearity, I'm having an issue with the whole battle system. I don't like the fact it took away the control that FFXII gave me and made two of the three characters completely automated.