Unfortunately, we don't really know exactly why and when it was first used to describe ammunition. In the late 1700s, "magnum" was first used to name extra-large bottles of wine, and "magnum opus" was used to denote a composer's best work. Things sort of progressed naturally from there, until magnum meant "bigger and better."
Does this mean that any cartridge bearing the name "magnum" is big and powerful? Hardly. The term has been applied to rifle cartridges from .17 caliber (that's the size of a BB) to more than .50 caliber (that's a half-inch), as well as to even larger-diameter shotgun ammunition. As with wine and works of musical prowess, its meaning is relative. "Magnum" doesn't necessarily mean "biggest and best." It just means "bigger" and (arguably) "better."
Possibly the earliest use of "magnum" to name a cartridge came in the latter half of the 1800s, when the British applied it to massive cartridges such as the 500/450 Magnum Express. Supposedly, comparison of these large cartridge cases with previous smaller cases brought to mind the difference between standard wine bottles and magnum-sized bottles. Whatever the case, the magnum name was used, and has endured.
It is not necessarily a useful descriptive term, because it is so relative. For instance, the 22 Winchester Magnum Rimfire (22 mag) is indeed more powerful than the 22 Long Rifle, but is a wimp when compared with other, larger, cartridges that may not bear the magnum name.
In summary, "magnum" is a term more useful for marketing than for realistically describing a cartridge and its performance.
- Russ Chastain
It may also apply to ammo that fits the same gun, but is more powerful. For instance, magnum shotgun shells have more "oomph" than standard-velocity shotgun shells.