Green Beret is on the right track, but the fact of the matter is that there is no simple explanation for "Magnum" as it is used to describe cartridges, because it is one of those words that is ill defined and has little meaning.
For instance, if you follow Green Beret's discussion of the .357 Magnum and the .44 Magnum, you might think that the use of the word is related to chamber pressure, which, in the cases of these two cartridges, is in the 35,000 range. The 9mm Parabellum (sometimes called 9mm Luger) also operates in this range, yet is not referred to as a "magnum" cartridge.
Green Beret also says: "...or has greater powder volume than its progenitor cartridge." This comes a little bit closer to the meaning of the word (if, indeed, it can be said to have a meaning.)
Back in the black powder days, powerful cartridges were termed "Express Train" and then simply, "Express." (See this
link for a more detailed explanation.) Later, when the British began using Cordite (at that time, a dual based smokeless powder comprised of nitroglycerine and nitrocellulose.), this term was modified to "Nitro Express," which you are no doubt familiar with.
In the USA, the term "Magnum" was preferred when trying to advertise cartridges of supposedly high power. For instance, in the USA, the term ".375 H&H Magnum" is used to describe what the British call the ".375 Belted Rimless Nitro Express." Or, what in the USA is called the .300 H&H Magnum is called "Holland's Super .30." You'll note that it is a relative term, since "high power" assumes a comparison with something of lesser power. (However, it is seldom specified what the "high power" is in comparison with; such things are usually left to the buyer's imagination.)
Anyway, the term
magnum (in Latin) can mean something great or large, and is used to describe many different things. One of the ways "magnum" is used describes a large bottle of champagne. In this sense, large cases based on the parent .300/.375 H&H cases came to be called "Belted Magnum," or just "Magnum."
The silliness of this is evident: most of the cases (except, notably, the Weatherby ones) were shorter, at ~2.5", than their parent .300/.375 case. Many of the hillbillies here in the USA thought that the belt at the base of the case was the reason for the use of the word "magnum," since it supposedly reinforced the base of the case. However, the reason for the belt on these cases was as a method of headspace. You will want to recall that, when the H&H cases were introduced, most high power cartridges were rimmed, or "flanged" as the British say, for use in large double guns. Indeed, both the 300 H&H and the 375 H&H have their rimmed equivalents for use in doubles. Because both of these cartridges have a very slight shoulder, they don't reliably headspace on the shoulder. Also, a rimmed cartridge often times has feed problems through bolt action rifles, especially the M98 based Mauser actions that were popular at the time, so the belted case was a form of a semi-rimmed cartridge that would function through a bolt action rifle, yet not require a distinct shoulder on which to set headspace in the chamber. The belt was, then, for headspace -- it had nothing to do with the cartridge's power. Most of the cartridges derived from the .300/.375 H&H case didn't require the belt, as they had sufficient shoulder on which to headspace -- the .458 Winchester Magnum being an exception.
Going back to the 9mm Parabellum issue I raised at the beginning, the Continental Europeans did not normally use the "Nitro Express" notation of the British or the "Magnum" issue of the Americans to denote high power cartridges.
Another note regarding chamber pressures: at one time, chamber pressures were measured by the "crusher" method. In this process, a copper pellet was placed in a hole in the chamber and sealed. When the round inside the chamber was fired, the copper pellet was crushed a certain amount, depending on the pressure, and the copper pellet was removed and measured, to see how much it was crushed. The amount the copper pellet was crushed was translated into a pressure measurement. Sometimes this is erroneously considered to mean the same thing as "psi" (pounds per square inch), however, the original measurement was in "cup" (copper units of pressure).
A similar method for assessing shotgun chamber pressure uses a lead pellet, so these tests reported shotgun chamber pressures in "lup" (lead units of pressure).
Later, chamber pressures were measured with piezoelectric crystals. This kind of crystal, like a Rochelle Salt crystal
Rochelle Salt, will generate electricity when pressure is applied to it, i.e., it is squeezed. In former days, this was the principle that phonograph cartridges worked (if anyone remembers such things).
Chamber pressures measured with the piezoelectric method are reported in psi, or pounds per square inch.
Also, a formula was developed to convert cup to psi, but this is not equal to the psi measurement obtained by the piezoelectric method.
So, it is possible for a cartridge to have chamber pressure reported in cup, or psi, or cup converted to psi. For the same cartridge, none of these figures will be the same... Even if the same cartridge is tested by cup and the cup is converted to psi by formula, this will not necessarily give results comparable to a psi measurement obtained by the piezoelectric measurement.
Beware, then, when comparing chamber pressures. Many folks, feeling that staring at a number on the page indicates an absolute comparison point to any other numbers, have gotten into no end of argument regarding this topic.
However, returning to the original point of the question, "Magnum," like "Express" and "Nitro Express," is a marketing term, and thus has a subjective meaning to each individual who succumbs to the marketing or who accepts such nomenclature as fact. The real truth is, one must examine the performance of each cartridge to get an accurate indication of its performance. Marketing names, such as "Magnum," are not especially useful for comparison. However, they are often used in the name of the cartridge, and therefore do have value as an identifier.
For instance, in this country, we had cars made by General Motors under the badge of "Saturn." Saturn offered a model called the "Ion." "Saturn Ion," therefore, was a useful term in describing to others what kind of car was being discussed, but the name had little value for describing how many electrons were actually in the car.
PS: You might also appreciate this fellow's description of another way that is used to measure chamber pressure:
Chamber Pressure Measurement
And as a final thought with regard to chamber pressures, I would note that the published chamber pressure information of cartridges like the .270 Winchester are as high as any magnum cartridge, when tested by the same means. So, in rifles, "magnum" does not necessarily mean higher operating pressures.
