This is a nice overview of common types but I still don't understand the "why" of some of their names, such as the difference between a .38 and a .380 (It's the same number!)
Because there's a lot more than caliber that matters. If you think about it, a .45 ACP is close in caliber to a .50 BMG. But there's hella powder behind a .50 and not so much behind a .45. The bullets themselves make a big difference also, there are at least a half dozen you can read more about here: https://gunvault.com/types-of-ammunition/
If you think having 2 cartridges that both have .38" caliber bullets is confusing, take a look at all the 9mm ones. Just in the ones that are actually called 9mm and not just caliber equivalent, there's 9x19, 9x25 mauser, 9x57 mauser, 9x39, 9mm winmag, 9x18 makarov, and a ton more I can't remember. Some of those are rifle cartridges, some handgun. They have different bullets with different characteristics and different bullet lengths and different bullet weights. And the shape of the cartridges behind them, the guns they fire from, and the amount of propellant each one has is all different.
the full name of .38 is .38 Special, and the full name of .380 is .380 ACP. IIRC, .380 ACP was so named to distinguish it from an earlier related design, .38 ACP. In common use, if somebody says .38 they're referring to .38 Special as that's by far the most common cartridge with .38 in it, and if they say .380 they're definitely referring to .380 ACP, though it's also sometimes referred to as 9mm Browning.
ETA: and in the case of both these rounds, .38" is actually the approximate diameter of the _case_ not the bullet; the diameter of .380 ACP bullet is 9mm, and the diameter of .38 Special is .357".
If you think having 2 cartridges that both have .38" caliber bullets is confusing, take a look at all the 9mm ones. Just in the ones that are actually called 9mm and not just caliber equivalent, there's 9x19, 9x25 mauser, 9x57 mauser, 9x39, 9mm winmag, 9x18 makarov, and a ton more I can't remember. Some of those are rifle cartridges, some handgun. They have different bullets with different characteristics and different bullet lengths and different bullet weights. And the shape of the cartridges behind them, the guns they fire from, and the amount of propellant each one has is all different.