I don't know about everyone else, but for me it's all about the storyline first and foremost. Do you get a true sense for who the characters are? What they're fighting for?.. and against? When a game makes me feel a connection with the characters and what they're going through, THAT'S a great game. All the extras like chocobo racing/ breeding, snowboarding.... they're just that...EXTRAS. Even the gameplay (the fight engine, how/ what kinds of magic are available....) and graphics are an after thought to me when compared to the quality of the storyline and character development.
I have only played (over and over and OVER again ;) ) FF4, 6, 7, 8, 9 and Tactics, but one thing I must say is that Square has never let me down when it comes to the storyline aspects. And I especially LOVE the music in these games. It is beautiful at times, adrenaline pumping at others...it ALWAYS aids in taking the player on the journey and making a deeper connection.
Along those lines, FF8 is in many ways my favorite game of the series. I think that everything is set up just about perfectly to play out the love story of Squall and Rinoa. The game is such that you as the player are sitting there on several occasions thinking "just tell her already," sitting on the edge of your seat in anticipation of what will happen next. And do I even have to mention how seeing Rinoa fall into Squall's arms after being rescued from the chamber gives you goosebumps?! There are SOOO many amazing scenes in this game that truly captivate the player, and so many intricately tied together storylines that make for one epic tale... how can anyone NOT like it?... Geez, all of this reminiscing... I may have to go dust off the Playstation again... :)