The things that I have always enjoyed about Final Fantasy is such a massive list, but I'll try to keep it short XD
One of them being the handling of the characters, their progression through the story and how they develop along the way. Normally it's for the better, which I kinda wish they had some to the bad side development, they tend to not get it right when they try, but points for trying! The characters are also colorful, they normally with astectially interesting designs (your mileage may very there though, I personally thought FFXIIIs character design was ridiculously over the top. I like the KISS principle when it comes it designing a character)
The characters also tend to have nice personalities, making me want to see them succeed and to keep playing to see how they react to the world and situations around them.
I also enjoy the battle systems in Final Fantasy. Some more then others, but I do like ATB element or the turn base, mainly the ATB system in the earlier games is ones I like. It makes the battles feel a bit quicker and yet demand an element of strategy of timing and hoping you got a healing spell before the boss gets a group attack out. I didn't care too much for FFXs, being turn based and all but the leveling up was rather fun even if it was just on a line. It certainly was pretty to look at.
The worlds that the game is based in as well is always something interesting, the way they present the world to the character is often large and granduous. Though it can feel a bit empty with how far towns can be from each other, it makes it hard to believe that the towns interact in trade or news at all, a small detail but I think too hard on that sort of stuff XD
Hmmm the stories are simple, starting off small and grow into something larger with the exception of a few of the games which didn't slowly swell so it felt a bit hard to get into the story. Or they also got a bit congested with confusing terms and badguys reasoning for what they're doing, which needs a bit of a narrative clean up and it can be damaging to the all around plot. But most of the time they handle the story okay throughout the game, UNLESS they do the 'read these massive paragraphs to get the plot!' No no. Don't do that FF, you have done better then that.
I think there is more, I do find the monsters fascinating (the older ones, the newer ones tend to look a bit too... shiny roboticy with less interesting organics. But it made sense for the setting of the game)
OH, the music is simply nostagic and very well done. They have interesting swells and builds the way the music goes, and the way they use the music in the games fit perfectly. Except for FFXIII, I don't think I have hated music in a game more then that game and it's sequel. Yeahhh lets toss in random vocals! That isn't jarring or annoying to listen to when you are listening to the characters talking at all!
That's about it for me, sure there's more but I think that massive post is enough XD