There's a lot of overlap in what can get called what but in general an arquebus is one of the first or so generations of handheld firearms. In form they usually bear a lot of similarity to a crossbow, including the long lever trigger. An arquebus often had a hook or other attachment point for being able to steady it on a stand (a kind of monopod) or a fixed emplacement on a battlement or what-have-you. The name derives from the dutch terminology for "hook gun" (Haakbus). A caliver is simply a kind of arquebus, typically one that has been standardized for service so that it is all the same caliber (hence the name). It's easy to forget how comparatively crude the earliest firearms were, but a typical arquebus might require slightly different projectile sizes than another, which as you can imagine is a bit of a logistical and tactical nightmare.

Originally the "musket" was just a larger arquebus, capable of penetrating heavy plate armor. But over time it eventually became a catch-all term for almost any flint-lock (or related) fired long-gun, especially as the match-lock of the original arquebus gave way to the snaphance, wheel lock, and then the flint-lock and the trigger transformed from the long lever parallel to the body of the weapon to a smaller mostly horizontally actuated lever more similar to modern firearms.

Musketoons were shorter barreled muskets (equivalent to a carbine today). Keep in mind that a typical musket would be nearly 4 feet long, making it unsuitable for some uses. "Dragoons", cavalry, or other mounted infantry might carry a musketoon or a shorter barreled firearm so that it could be more easily handled and fired on horseback. A fusil was a type of flintlock musket carried by "fusiliers" who would guard and escort artillery. These were sometimes shorter barreled than other muskets, sometimes not, initially this was a special role because the flintlock was not yet ubiquitous and it was desirable to separate forces that used matches for firing away from the large amounts of exposed gunpowder used in artillery. Over time the term (and the firearm) lost its unique meaning and many ordinary infantry units armed with the same sort of muskets that everybody else had were called "fusiliers".