Main Characters:
Zidane: a lively little monkey-boy aged 17 that is a thief and something of an actor. Like all good main characters, he's got a strange mottled past.

Garnet/Dagger:  She is the daughter of the Queen, making her a Princess.  Without giving away too much of the story, she desires to be kidnapped and runs away with Zidane to have some fun adventures.

Steiner:  a older goofy knight that gets dragged along with Garnet and Zidane. He hates the boy and hates all the passes that Zidane makes at Garnet. He gets hopping mad and thinks she deserves more.

Amarant:  A strange man/creature that is very aloof and joins and leaves your party on a whim. He's helpful -- when he's around.

The Cook [whose name I forgot]:  He's very freaky, nothing even humanoid looking and he can eat the monsters. c_C

Eiko: is a very cute little 8-year-old that fights by playing her flute.  She's got a crush on Zidane.

Freya:  A very interesting female animorphic character.  She's quiet, intellegent and reserved.

Vivi:  A mage of sorts, he has glowing yellow eyes under a hat of all black.  Despite that he's very clumsy.  But very lovable.

Garland:  Hard to describe without spoilers, but he's sort of a creator.  An older gentlemen with ties to both Kuja and Zidane.

Kuja:  One of the most girly men you'll find.  He's the villain of the game, wearing not much at all, and having a grudge against the world.
 

Story background:
What starts off for Zidane as a regular theifing party ends up as a full scale kidnapping as the Princess Garnet ends running away with Zidane and his gang. Consequently, their ship is destroyed.  Through several trials they meet up with a host of interesting companions and go about trying to discover the source of the mists that cover the lands.  They also meet Vivi and try to discover more of him and his past.  But they meet up with Kuja and things take a turn for the worse as they're pitted against Black Mages and their very existences are at sake.  Does Zidane really want to learn who is he?  Can Garnet learn to cope with her past -- and her future?
 

Slash potential:
Not nearly as much as FFVII or FFVIII

KujaxZidane:  This is the most popular, and pretty much the only one I see at all.  Sadly, it doesn't really blossom until the very end of the game.  But it's good enough to make the game good.  Sadly, it's the only good yaoi pairing in this game.

GarlandxKuja:  Because... X_x;;; yeah. Well, twisted people could think of it.

AmarantxZidane:  It's possible, but I don't wanna read it.

Why?

Well, Zidane more or less leaves everything and everyone he knows and loves to go back for a "presumed" dead Kuja and risks his own life and more or less kills himself [although he survive, so we assume] for Kuja.  *_* It's one of those really neat redeeming moments.  The end of the game is awesome.
 

Links:
I haven't heard of a single FF9 fanfiction yet, but you can see a rather nice page dedicated to Kuja here:
http://yatenkou.virtualave.net/angel/