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.
Berserk Button: Do not harm his friends. He will not take kindly to this.
Hollywood Mid Life Crisis: Another theory behind Jack's lost love for Halloween and being so enamored by Christmas. After his attempts at running Christmas go bad, he rediscovers his love for Halloween.
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 kidnapping 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? I knew you would be. You don't have to worry about another Christmas this year.
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)
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!!!"
The Leader: Type IV. Due to his skill and charisma, all of Halloweentown loves him and would follow any Zany Scheme he's concocted.
Loves the Sound of Screaming: A rare heroic example. 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 (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.
Modest Royalty: In terms of appearance, as the Pumpkin King wears no crown. He's certainly not modest in personality.
Mood-Swinger: Jack frequently and quickly swings between elation, depression, and anger. Many fans speculate that he has a mood disorder (bipolar disorder is the popular choice, but that's not actually how it works, most of the time).
Narcissist: He and the whole film revolve around his bull-headed attitude, and attempt to steal a holiday that doesn't belong to him.
Obliviously Evil: Jack means no harm at all and doesn't have a malicious bone in his body, but it's his actions that are creating nearly every problem in the film. Thankfully, he does eventually come to his senses.
Oblivious to Love: Throughout most of the movie, he only views Sally as a friend, and has no idea that she's in love with him. He catches on eventually though.
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.
Unrequited Tragic Maiden: Seems to think she's this, as evidenced by her song, where she openly says she isn't "the one" for Jack. However, it's proven wrong when Jack does reciprocate her feelings after all.
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.
Belligerent Sexual Tension: One interpretation of Lock and Shock's relationship; Fanon says have a mutual crush on each other, but express it by teasing and yelling at each other.
Butt Monkey: Barrel is constantly at the receiving end of pranks and smacks from his cohorts. (Just watch "Kidnap the Sandy Claws".) But usually when he pulls a prank on Lock or Shock he gets away with it.
Big Bad: Though all the citizens of Halloween Town dedicate their lives to scaring people, they do it purely for laughs and are mostly harmless. Oogie is the only really EVIL person there, and is implied to have actually killed people.
Blatant Lies: In an interview in his biography, Tim Burton described Oogie as "A nasty neighbor", said that there weren't any true villains; ohh yeeaaah... Anyone who's been banished, has tried to eat Santa Claus, and tried to forcibly take over all 7 holidays musn't be that evil!
Boisterous Bruiser: A rare evil example. He loves singing and dancing...while he horribly tortures his victims and later cook'em and eat'em.
Dirty Coward: Despite being a hundred times bigger than Jack (at least in terms of sheer bulk), he whimpers and runs away if he even thinks Jack has the upper hand. Considering he's just a big burlap sack full of bugs, this behavior is reasonably justified.
The Dreaded: Even the other residents of Halloween Town are implied to be terrified of him.
Evil Counterpart: He's this to Jack- he's just as terrifying (if not moreso), but unlike Jack, he's more of a sociopathic monster than a prankster.
Fat Bastard: Not only is he a big, nasty bag of bugs with a love of torture and casual cannibalism, he cheats at dice. In fact, Oogie Boogie was the biggest, heaviest puppet in the entire cast, needing the tightest joints in his armature skeleton for proper movement.
Faux Affably Evil: He may look like an overstuffed potato sack living in a cross between a cheesy blacklight funhouse and a casino, but he's actually a sadistic monster who slowly torments people with rigged games of chance before cooking them up and eating them.
For the Evulz: The only Halloween Town resident who is truly evil.
For some of them, anyway, at least onscreen. If you pay attention, the kid with the shrunken head is named Timmy (it says so on his stocking behind him), and in the book, The Nightmare Before Christmas: The Film, The Art, The Vision, on some of the concept art, it shows the kid who gets the big snake is referred to as "The Infant."
Actually, if you look at the stockings on the fireplace behind him, his name is Timmy. It probably falls under Freeze-Frame Bonus.
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. He actually was scared, but was to shocked to properly express himself