The bugs are Oogie's army. They were the original inhabitants of Bug Day and when it was forgotten they immigrated to Halloween Town with their king. They are under his order to take over the town and turn it into the New Bug Day.
The bugs are giant sentient bugs that attack with all kinds of techniques and weapons. They are based off of real bugs such as caterpillars, spiders, flies, and centipedes. They are apparently organized, as they have ranks with the boss bugs acting as generals over the lower ranking soldier bugs. Some have very silly natures, like one kind of bug that pushes another around in a stroller. Others are more threatening in nature but all are loyal to Oogie and are seeking to restore their holiday of bug day to its former glory. Though the army is mostly made of bugs, there are a few non-bug members (this refers to characters other than Lock, Shock, and Barrel) and there are many animated cow skeletons, trees, headless horseman skeletons, and a giant snake. They seek to conquer the inhabitants of Halloween Town but appear to be smart enough to distinguish friend from foe as they do not attack Lock, Shock, and Barrel. They are sapient and utilize a wide variety of weapons in the fight against Jack Skellington. Some of the boss bugs show even great displays of intelligence such as steering Jack towards them with signs, taking hostages and attacking Jack while he was carrying an exploding mushroom. They came close to realizing their goal of total domination, however, the tide was turned by the Pumpkin King and were all destroyed except for Oogie Boogie, who vanished into his hideout.
This section details the boss/generals in the Bug Army.
1. Mega Spider
The spider is first brought to Jack’s attention by the Mayor, who tells Jack that it has taken over the city hall. To make sure Jack finds his way into his trap, he has one of his minions stand out in front with a sign. Inside, City Hall is covered in webs, which he uses to fight Jack.
2. Giant snake
The giant snake is encountered in Dr. Finkelstein's lab, where it has taken the doctor hostage. At first, Jack thinks the snake is the doctor's pet, but the doctor angrily yells otherwise. The snake then stretches itself to surround Jack and attack him by biting at Jack, spitting purple balls of fire, and summing beats to protect its head. Upon defeating the snake, it's knocked out and left at the mercy of Dr. Finklestein. It is the only boss besides Oogie to service the game and actively play a minor role in The Nightmare Before Christmas.
3. Rolly Polly/Enormous Polly
The big centipede is the boss of the pumpkin patch. It appears to be very smart until Jack has the exploding mushroom and, like his spider friend before him, has a bug point the way so Jack doesn't miss his little showdown. At first, he seems to be a large centipede just a little bigger than Jack. It can also have fire exploding bug shells and you must use the exploding pumpkins scattered about to knock it on its back, making it vulnerable. It will shortly be defeated after where it splits open revealing a giant centipede. The bug itself is just a centipede shaped sack like Oogie Boogie but still a giant centipede that will pop out and throw exploding centipede shells. The bomb you're carrying is with a timer that's counting down and every hit you take speeds it up. If it hits zero you lose. The centipede puts up fire resistents but it's not enough to defeat Jack who delivers the musher to the big witch in time to get the pumpkin king buster.
4. Colossal Moth
The moth sets itself up in a base under the cemetery, then sends its minions to set a trap for Jack to take control of Dr. Finkelstein's lab, sabotage the lab's security system and take Igor hostage. Despite being cut in several cage traps Jack managed to reclaim the lab and free Igor. After that Dr. Finkelstein directs Jack to look closely out back there, the boss has a single bug direct Jack to his lair in hopes of eliminating him once and for all. The moth's main method of attack is to fly up and drop down on Jack but he can also spit poisonous clouds that follow Jack and drop several round pellets. If the moth takes damage it gets more aggressive, firing multiple poisonous clouds at once. After enough damage is dealt the moth will start to bleed and fly higher up to the top of the chamber where acidic blood floods the chamber. It's a hard fight but the moth falls to the power of the pumpkin king.