Read, Read, and Read some more.
There are four basic ways that most, but not all, cartridges are named as far as the number is concerned.
#1 is the groove diameter of the barrel/ the actual diameter of the bullet. The groove diameter is the distance across the inside of the barrel that is the widest. An example would be the .308 Winchester. The groove diameter as well as the bullet diameter is .308 thousandths of an inch.
#2 is the bore diameter. Bore diameter is the distance from rifleing to rifleing or another way to put it is the size of the hole in the barrel
before the rifleing is cut. This is the shortest distance from one side of the inside of the barrel to the other. An example is a 300 magnum. While the 300 uses the same bullet as the .308 Winchester it is named a 300 because each side of the rifleing is 4 thousandths of an inch tall. Double this number for each side of the barrel and you get 8 thousandths. Subtract 8 thousandths from 308 thousandths and you get 300 thousandths of an inch.
#3. They pull a number out of the sky. An exampole of this would be a 44 Remington Magnum. If one didn't know you would guess that the bullet diameter would be somewhere in the neighborhood of 440 thousandths of an inch but in fact it is only .429 thousandths of an inch. So it should really be called a 42 or 43 magnum. But I guess that doesn't sound as cool as 44 magnum.
#4 is cartridges like the 30-06. 30 meaning 30 caliber and 06 meaning the year that it was adopted in its final form... 1906.
The names associated with the cartridges are......
If it is a manufacturer the they are usually the ones responsible for commercializing the cartridge and possibly inventing it. Such as the .308 Winchester, 280 Remington, 40 S&W, .357 Sig, .480 Ruger etc...
If a persons name is attached then that person is usually the one responsible for actually inventing or creating it before it was ever commercialized. Such as the 416 Rigby, 257 Roberts, 7-30 Waters, 35 Whelen, etc...