Republic
Trooper: A bit dull and straightforward of a soldier story at times, with choices ranging from "bleeding-heart patriot" to "bleeding-heart patriot who's kinda ruthless". Decent companions and side characters, though.
Jedi Knight: The classic. The KOTOR III. This is the "main" story of the eight, directly involving the big bad Sith Emperor. You play as the heroic hero going on a hero's journey to save the galaxy from a very unheroic dude with a lot of heroic (and some unheroic) friends at your side. Perhaps a bit straightforward, but it's pretty solid.
Smuggler: Aside from Consular, most stories have a pretty rocking Act I but kinda less epic Acts II and III. Smuggler seems to be the biggest victim of this, with the later story making feel a bit too chained up by the Republic; then again, perhaps Act I's freelance treasure-hunting story was just more fun than usual. However, the banter and humor is pretty solid throughout; it's probably the funniest story no matter what dialogue you pick, and that makes it worth it. Some good companions and side characters too.
Jedi Consular: This one's...meh. I don't think it's as bad as some others say it is, but the Prologue and Act I is kinda dull. The equally mediocre companions don't really help brighten the mood. In general it's a bit more slow-paced; you're a Jedi Consular delving into the mysteries of the Force while trying to hold the political fabric of the Republic together from behind-the-scenes.
Empire:
Bounty Hunter: The Imperial version of the Smuggler story, with similar pros/cons. Some good companions (aside from Skadge...most people hate that guy), and it can be fun to play as the wise-cracking bounty hunter hunting down big Republic game.
Sith Warrior: The dark side mirror of Jedi Knight, though with significantly less of that "main storyline" feel. I still enjoyed playing as an unstoppable Sith badass, and it has some great companions and solid villains.
Imperial Agent: My favorite story, and from what I've seen one that is usually considered the best. Choices actually matter (at least more than in most other class stories), and there's plenty of interesting turns and hard decisions to make that can open/close new paths later on. Decent villains and side characters, though I found the companions to be...a mixed bag.
Sith Inquisitor: It's a bit of a debate, but I honestly thought Sith Warrior had a better story. SI was sold as the Palpatine kinda story, all about manipulating others and realpolitiking your way up the Sith ladder to the Dark Council. In reality, though, while it does have the slave-to-master story (you literally start as a slave), it's more about zapping people for shits and giggles and playing Ghostbusters to collect enough power to just brute force your way to the top. There's just moments where it feels you have no say in your character just walking into an obvious trap, and you end up having to get rescued by your companions or some other NPCs. It's a bit sad for me that the Warrior story seems to give me greater freedom in manipulating others, in rallying allies, etc. Companions are good, but the Act II-III villain is some weak stuff.