jimpage and aguila blanca both did a pretty thorough job of explaining everything. one that I haven't seen brought up yet however is 9MM NATO. there is some ammo floating out there marked NATO. IIRC its normally 124gr and loaded to near +p power but holds all the same dimensions as 9 para/9x19/luger.
not long ago I started picking up my 9mm brass so incase I ever decide to start reloading and I found one case that was marked 380. I about pooped a brick because for the most part I've been using once fired ammo I don't know which of my 3 9mms I fired 380 out of but then the scarier question occurred to me. was that loaded as 380 or was it loaded as 9mm?
since then I've started looking at my ammo very closely before I load it.
also IIRC 45-70 denotes the amount of black powder charge for a round. even though the 45-70 was invented after smokeless powder became the norm, they still loaded it to the same power that you would expect from 70 grains of black powder. there is also 45-80, 45-90, 45-100 and 50-90 calibers that are all named for their diameter in inches and the black powder charge(even though they are not actually black powder).