The trebuchet (sometimes "trebucket", but that is less common) is essentially a lever supported by an axle through upright posts of a frame, closer to one end than the other. The long end has a sling resembling a handheld sling and a release pin. The short end has the counterweight, which is many times more massive than the mass of the object to be thrown.
The objective is to apply the energy of a large mass slowly falling for a short distance into throwing a relatively small mass at high velocity for a long distance.
The two main counterweight designs are "fixed" and "hinged". The fixed type has the weight solidly fastened to the end somewhat like a hammerhead. The hinged type suspends the counterweight from short vertical bars that are hinged at the short end of the lever and (preferably) at the point where they attach to the counterweight itself. The advantage of the hinge suspension is that as the counterweight drops during the firing sequence, it follows a more nearly vertical path rather than an arc.
A fixed counterweight trebuchet may be equipped with wheels which will allow the entire engine to move first backwards and then forward as the counterweight end drops, which again allows a more nearly vertical drop.
A third type, the "traction trebuchet" does not have a counterweight, as such, on the short end. Instead, it has one or more ropes which a crew may pull down. In effect, the crew acts as the counterweight. More frequently these days, this type is being called a "mangonel" by SCA siege operators to distinguish it from the counterweight type, although mangonel is sometimes used to mean an onager. The word "mangonel" itself is more of a generic term for several types of siege engines.
How they work
The long end of the trebuchet is equipped with a short pin (resembling a headless spike bent slightly forward) on its tip , and a rope sling with a shallow pouch in the middle. One end of the sling is permanently fixed near the long end of the lever, and the other end has a loop or metal ring which is simply slipped over the pin. In preparation for firing, the long end of the lever is hauled down to near the uprange (rear) end of the frame. The sling is laid out along a trough, with the pouch and its ammunition resting somewhere under the pivot point or the counterweight. Some kind of trigger mechanism holds the lever down until it is ready to be fired
When the trigger is released the counterweight drops, forcing the long end to swing up rapidly. The sling is pulled back along the trough and the pouch begins to swing out backwards and up. At approximately 45 degrees from the horizontal, (when the lever has travelled about 90 degrees) the loose end of the sling slips off the pin, allowing the ammunition to slip out of the pouch and fly off downrange.
A trebuchet can be extremely accurate (particularly the fixed and hinged counterweight types) because successive shots can land in a relatively small grouping, and one can make adjustments to either the pin angle or the sling length or both to obtain better results, which may then be repeated with consistency provided the ammunition is consistent in both mass and volume.
Trebuchets can be scaled up in size for throwing very large projectiles, more easily than onagers, ballistae, or arbelests. Hence the nickname of "the atomic bomb of the Middle Ages".
Traction trebuchets have a very short reloading time because you do not need to winch the arm down to prepare them for firing. For SCA combat, this allows a rapid rate of fire.
They cannot easily be used in a direct-fire manner (against, for example, massed troops), as can arbalests or ballistae.
The massive counterweights make them less portable on a battlefield, and more difficult to aim left or right. Historically, trebuchets were more likely to be constructed on the spot in anticipation of a lengthy siege. The defenders might even surrender once they've determined what was being built.
While the lack of a massive counterweight makes them more portable and easy to aim left or right rapidly, the range accuracy of traction trebuchets suffer because it is not easy for the crew to repeat exactly the speed and strength of the pull from one shot to the next.