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A trebuchet was a heavy siege engine used by the Broddring Empire. A long wooden arm with a basket at the end was fastened with a system of ropes and gears to a strong wooden platform, which was mounted on wheels, enabling the trebuchet to shoot a projectile, such as a barrel of liquid fire, placed in the basket a long distance. When a certain rope was undone, the trebuchet's wooden arm would hurl the barrel into the oncoming ranks of the enemy soldiers.
The Broddring Empire's trebuchets caused terrific damage to the Varden's army during the Battle of The Burning Plains. However, Eragon managed to destroy many of them by taking over the minds of the guards standing near the trebuchets and forcing them to hack through the ropes which held the trebuchets' arms in place.
Please note that this section is based on the real-world machine called the trebuchet. In Inheritance, no technical description beyond that cited above is given.
The trebuchet is the heaviest of the early-medieval siege engines. Technically a class of catapult, trebuchet differs from most other catapults, which use torsion (i.e. a wound rope, much like an oversized crossbow), in that it uses the leverage principle to launch the projectile. The mechanism consists of a long, usually wooden, arm mounted on an axle in a wooden frame. On the shorter end of the arm a counterweight is fixed, on the longer, a sling that holds the projectile. The weapon must be "drawn" to fire, i.e. the counterweight lifted from the ground. Once it is released, the machine works like a two-armed (first-class) lever, using the mechanical advantage principle with the counterweight arm being the shorter one, which enables the machine to give the projectile considerable speed. Firing angle can be controlled by releasing the projectile from the sling (done through a rope or pulley) at the desired moment. As the machine is in most cases several meters tall and quite heavy, the frame is usually fixed to a mobile wheeled platform.
Trebuchets were much more accurate and powerful than torsion catapults like the ballista or the onager (which is the engine most often identified with the more general term "catapult"). While ballistae were quite easy to aim, their projectiles (usually spear-sized javelins) are inefficient against buildings and walls. Onagers, on the other hand, virtually could not be aimed. Also, one might build a trebuchet as large as the construction material will support. An iron machine of this design could be indeed enormous, although none have ever been created. Still, historical trebuchets could reportedly hurl stones as heavy as a ton for distances of dozens of meters and could achieve a range of up to 300 meters with projectiles weighing around a metric cent (100kg). Also, they are capable of quite rapid fire - if properly manned, a good trebuchet can fire up to two times a minute, faster than most torsion catapults. By its size and the size of the projectiles, trebuchets are ideal against heavy fortifications.
Although trebuchets have greater power and range than catapults, they also have drawbacks. Trebuchets are more complicated than torsion catapults and therefore are plagued by mechanical breakdowns, require more maintenance, more skill to operate and a larger crew. They are also less mobile and harder to transport, not to mention more expensive to build. Finally, while they may be used against infantry, it is usually considered overkill and a ballista may do the job better, considering that for the resources it takes to field a single trebuchet, several ballistae could be deployed. Also, trebuchet projectiles are easy to evade for individual fighters (as long as they see the projectile early enough).