A caliber is one onehundreth of an inch. 25 caliber is a 1/4 inch and 50 caliber is 1/2 inch. As far as the names go there is no one system that applies to all of them.
Here are some explainations of cartridge names.
45-70, 45 caliber and 70 grains of blackpowder
44-40, 44 caliber and 40 grains of blackpowder
38-40, 40 caliber and 38 grains of blackpowder
30-30, 30 caliber (308) and 30 grains of smokeless powder.
30-06, 30 caliber (308) and improved in 1906 (introduced in 1903)
308 Win, 30 caliber and the same case family as the 30-06, but shorter.
243 Win, 308 case necked down to .243.
358 Win, 308 case necked up to .358.
7mm/08, 308 case necked down to .284
35 Whelen, 30-06 necked up to 358 caliber and named after Whelen.
270 Win., 30-06 necked down to .277.
250/3000 or 250 Savage, Same case family as the 308 & 30-06 but shorter yet than the 308 and is in 257 caliber. The 3000 means it will push a bullet (87grain) to 3000 FPS which was a big deal when it was introduced.
22/250, It is the 250/3000 necked down to .224
Then there is all the metric ones, all the magnums both belted and beltless, and all the wildcats.
Hope this helps.
Hud