Zero is entirely loyal to his master. Despite his ghostly form, he acts very much like a normal dog; his favorite activity is fetching the bones pulled from Jack's skeletal body.
Despite Zero's loyalty, he (along with Sally) was the only Halloween Town resident to show reservation with Jack's unhealthy obsession with Christmas. It would appear that Zero is rather perceptive, as his concerns proved to be valid when Jack's Christmas turned out to be a disaster.
Zero's primary concern is Jack's happiness. Despite his reservations with the Christmas ordeal, Zero aided Jack with overtaking the holiday with the hope of helping the latter live out his dream. He did, however, try to steer Jack in the right direction by reminding him of his true purpose (doing so by showing a portrait depicting Jack with a pumpkin). Like Sally, Zero's efforts fell on deaf ears until the third act of the movie.
Zero is a ghostly dog that appears to be made of a white sheet, much like a stereotypical ghost. Like Pluto, he has basset hound-esque floppy ears. His "eyes" are just empty holes in the sheet. Zero's snout is very long and thin, and his upper jaw curls upward at the tip. His nose is a small glowing jack-o-lantern like Rudolph's red nose, and his collar is a faded brown color.
Zero first appears when he is summoned from his grave by Jack after the yearly Halloween celebration ends and comforts his master as the latter expresses his grievances over Halloween's waning excitement. Zero later accompanies Jack for his walk through the woods, and in during which, begs to play fetch. The two later find a mysterious area filled with multiple doors that lead to similar worlds focused on different holidays. Jack enters the Christmas door, and Zero stays behind. When Jack returns, Zero is by his side as the former enters Halloween Town's square with various Christmas-related items. Jack makes it his mission to study Christmas and incorporate its customs into Halloween Town. Zero becomes increasingly wary of Jack's growing obsession, and when Christmas finally comes (which Jack forcefully decides to take over as Santa Claus for the year) Zero is selected to lead the sleigh so that his shining nose can guide the other reindeer into the human world. During the escapade, Jack and Zero are shot down from the sky by the military, though both survive. In a graveyard, Jack comes to terms with his role as the Pumpkin King, much to Zero's delight, and the two return to Halloween Town in haste to rescue Santa Claus from Oogie Boogie. During the battle, Zero barks aggressively at Oogie and looks in shock as the villain dies at Jack's hands. During the finale, Jack and Sally confess their love for one another, and as they do so on the spiraling hill, Zero watches on before flying into the air and turning into a star.
An emoticon version of Zero appeared in the As Told by Emoji episode based on The Nightmare Before Christmas. He started the short as an emoji of a normal dog (alongside an emoji of a pumpkin) before transforming to his well-known ghostly appearance.
Zero appears as an ally to Jack, vital in certain parts of the storyline. After hearing from Corpse Kid and Clown that Oogie's baddies had kidnapped Sally and taken her to the cemetery, Jack rushes there himself. He finds the Hanging Tree, though his Hanging Men are missing, who states that he doesn't know where Sally is but adds that his men might. Jack states that he could track them down and ask, and mentions that he could use Zero's nose for it. Hanging Tree directs him to Zero's grave and, after a skirmish with Lock and some skeletons, Jack reaches the grave and calls Zero. He emerges and leads Jack to five separate places in the graveyard, where each Hanging Man is hiding.
He is seen again much later after Jack has all but undone everything Oogie did in Halloween Town. He learned from Oogie's shadow that the villain was currently in Christmas Town, so he calls Zero again from his grave to help him find the key to Spiral Hill so that he can reach the Hinterlands. Zero's nose again proves helpful, and Jack finds the key.
Zero's grave is accessible at any point once the Graveyard can be reached, but he will never be home except for those two times. Apart from that, he is only seen in the closing scene, as Jack and Sally share a heartfelt moment like the ending of the film.