That needs to be .300 Ultra Win Mag (instead of Win Ultra Mag) in my example BUT you still get the idea. Hope that helps explain it.
Actually, it's Remington Ultra Mag.
Basically, what InKed said covers it. "Magnum" has just become an adjective to describe increased performance. Over what exactly has become much less clear these days, with many cartridges that do no carry the magnum moniker equalling or exceeding the performance of the "magnums" (ex. .25WSSM vs. .25-06 Rem)
Virtually all cartridges have the name of the developer, the original weapon so chambered or the company that first picked them up in the name somewhere, but some are popular enough that there's little confusion when the pronoun is dropped.
"7mm Mag"-there are several, but the Remington mag is by far the most common, and usually assumed to be the cartridge expressed when there is no other name attached.
".300 Mag"-again, there are a number of them, but the .300 Winchester magnum is the one usually assumed with the abbreviated denotation
".357 mag" is technically .357 S&W Magnum, but there's really no need to denote S&W. Same with .41 and .44 Remington Magnum.
The common ones are:
Rem. Mag (Remington)
Win. Mag (Winchester)
S&W Mag (Smith & Wesson)
H&H mag (Holland and Holland)
Wthby Mag (Weatherby)
WSM (Winchester Short Magnum)
WSSM (Winchester Super Short Magnum)
RSAUM or SAUM (Remington Short Action Ultra Mag)
RUM ((Remington Ultra Mag)
There are others, like H&R, with only one or two cartridges:
.32 H&R Magnum (Harrington and Richardson)
.327 Federal Magnum
And sometimes no name:
10mm Magnum