The barrel holds the mainspring which drives the movement. The mainspring is hooked onto the central arbor that the barrel turns upon. Usually, there is a barrel cap that clips onto the barrel holding the mainspring in place. Sometimes on slim calibres no barrel cap is present and the movement of the mainspring is limited by the main-plate the barrel sits upon.
The barrel-arbor is the central axle inside the barrel upon which the mainspring hooks onto.
The click is a lever with a "beak" which is activated by a spring, engages with the teeth of the ratchet wheel, to allow it to turn in one direction only.
A fusee, the angle of the cone of the fusee, correlates to the change in force of the mainspring, to produce a more constant force transmission to the escapement, through the gear train.
The hook (s) is/are found at the end of the fusee chain
Ratchet wheel, a toothed wheel screwed onto a square hole on the barrel-arbor. A click prevents it from turning in the opposite direction for unwinding.
Stopwork,"Device comprising a finger-piece b fixed to the barrel-arbor and a small wheel a called the Maltese cross, which is mounted on the barrel-cover. This mechanism limits the extent to which the barrel can be wound." (Berner)