Cyclops are a race of mythological creatures known for their having one eye. One particularly famous cyclops is the one defeated by Odysseus in Homer's "Odyssey''.
The Cyclops
This particular Cyclops appears as a citizen of Halloween Town, first seen in the beginning of the movie and when the Wolfman asks him "Wasn't it terrifying?" he and Mr. Hyde reply "What a night!", with Hyde clutching a chain around him.
When looking for Jack Skellington, a Vampire says he looked behind the Cyclops' eye. Which might've been this Cyclops. When the Vampire states "This fog is as thick as," he says "Jellied brains." He is excited when Jack returns from Oogie's lair and is thrilled with Santa's gift of snow.
Suprised Cyclops