Similar to the Horseless Headless Horsemann from Mann Manor, Monoculus is a boss monster that attacks both teams trying to control the center point in Eyeaduct.
Monoculus' attacks consist of eye-missiles that shoot out from its pupil. These always deal critical hits. When it takes enough damage from critical hits Monoculus enters "rage" mode and adopts an angry squint. In this state its rate of attack, projectile speed, and accuracy are greatly increased.
Monoculus will randomly teleport around the map via a swirling blue portal. If players jump into this temporary portal, they will be transported to the Underworld. The underworld environment is hostile to players, dealing a slow but constant stream of damage to anyone within. Players must cross the short expanse of the Underworld to the exit portal which will spawn them back on the map proper with a temporary speed, damage, and invulnerability buff which allows them to deal even greater damage to Monoculus.
Upon defeat, Monoculus will spawn a special portal to Loot Island, a safe part of the Underworld where players can find the Bombinomicon which, when touched, transports players back to the map, unlocks the Bombinomicon item for them and grants the Dive into a Good Book achievement.
In the "Bombinomicon" comic, we learn about the origins of MONOCULUS! As a child, the RED Demoman went to the castle of a wizard named Merasmus to find a job on Halloween (he disappears from the Scottish moors the rest of the year) because his mother pushed him to. Young Tavish Degroot swept the floors of Merasmus' library all night until he came across a talking book called the Bombinomicon. Even though he was specifically instructed not to read it, the book enticed him with recipes for making bombs, and upon reading it, yanked his eye out of the socket and laughed. The eye floated every year across the moor on Halloween until one year, as the RED team was having a Halloween party, Merasmus showed up uninvited (turns out he's the RED Soldier's roommate). Merasmus and Soldier had an argument, and as revenge for snapping Merasmus' staff, both teams were forced to fight MONOCULUS!
(It should be noted that this is the second or third time Valve have offered an entirely different explanation for the loss of Demoman's eye.)