Badass Bookworm: He attempts to master the magic of Christmas through study and scientific research, and while his methods fail entirely, the logic he employs suggests a strange sort of genius at work.
Bad Santa: Though, in his case, it's less due to maliciousness than just not being very good at Santa's job.
Beneath the Mask: All of the Citizens of Halloween Town (except Sally) believe Jack to be bold, loud, and happy. Too bad they have no idea that underneath that bony exterior was a guy that was practically going through depression before he discovered Christmas Town.
I Have Many Names: Jack Skellington, the Pumpkin King, the King of Halloween, Bone Daddy, Skeleton Jack, Mr. Unlucky...
Idiot Hero: As detailed on the main page, he's not an Idiot Hero because of an inherent lack of intelligence; he's an Idiot Hero because he's never encountered non-Halloween stuff before and he's too excited to listen to common sense.
The Kindnapper: Jack Skellington has part of his being a Nightmare Fuel Station Attendant. His way of indulging himself in his new passion for Christmas not only involves planning to run the show himself, but kidnapping the one who already does to enable his doing so. And he sees it as a favor for Santa, too! Despite Jack's unquestionable status as The Hero, though, his kindnapping is clearly shown as not being a good thing, especially because of what it leads to...
Santa: (bursts out of the bag) Let me out! (the Halloween Town citizens gasp in awe)
Jack Skellington: Sandy Claws - in person. What a pleasure to meet you. (prepares to shake but then looks down when their HANDS touch)
Jack Skellington: Wh - ! Why, you have hands! You don't have claws at all!
Santa: (dazed) Where am I?
Jack Skellington: Surprised, are you? You don't have to worry about another Christmas this year.
Santa: Wh-what?
Jack Skellington: Consider this a vacation, Sandy. A reward. It's your turn to take it easy.
Santa: B-But there must be some mistake!
Jack Skellington: See that he's comfortable... (the minions start to close up the bag, but then stop) Just a second, fellas! Of course! That's what I'm missing! (takes Santa's hat)
Santa: B-But...
Jack Skellington: (as he's putting on the hat) Thanks.
Santa: Hang on - you just can't - (has the bag thrown over him again) - Hold on! Where are we going now?
Large Ham: More subtle than Oogie, but still. "I AM THE PUMPKIN HAM KING!!!"
To further explain, he, and most of Halloween Town, enjoy the sound of screaming not out of sadism but because it's their job which they enjoy.
Plus the screaming comes from harmless pranks rather than torture.
Made of Iron: In his first appearance he sets himself on fire (though only briefly), and then later in the movie he falls at least a mile, lands on a stone statue, and doesn't suffer any permanent damage.
Vague Age: In personality and voice, he comes across as being in his late twenties or early thirties, but he's an undead skeleton man, so this has no connection whatsoever to how long he's been around. And aside from being adult-sized it's impossible to gauge his age from his looks. He's probably older than thirtyish, though it wouldn't be impossible for him to be younger.
Affably Evil: They really are terrified of Oogie, but in their minds, helping him is the most fun thing they can think of.
Badass Normal: In any video game (sequel, prequel, or spin-off) they do things like control monsters, capture holiday leaders, fight the protagonists, and convince Maleficent herself to resurrect Oogie.
The Brute: Barrel. Switches to Lock in Oogie's Revenge
Butt Monkey: Barrel is constantly at the receiving end of pranks and smacks from his cohorts. But usually when he pulls a prank on Lock or Shock he gets away with it.
The Stoic: Not a straight example, but the kid does deserve credit for not screaming when he finds his gift, and not being bothered that his home's been invaded by an undead Santa Claus.