Bear in mind that cartridges are not always referred by the same name throughout history. For example, when Smith & Wesson produced their first revolver model, they made it fire a .22 cartridge, which was then referred to as a ".22" or a ".22 caliber" in 1857, since there were no other cartridges with the same caliber bullet. This cartridge contained 4 grains of black powder in it and a bullet weighing 29 grains (or 2 grams in the metric system) and measuring .222 inches in diameter. In 1871, Smith & Wesson produced another cartridge in .22 caliber diameter. This cartridge used the same 29 grain bullet as its predecessor, but the case was a bit longer than the older one and contained more black powder (5 grains), in order to produce extra power. Therefore, they began to refer to the old .22 cartridge as ".22 Short" and the new one as ".22 Long". Both these cartridges were used for revolvers. Soon afterwards, Remington, Stevens, Winchester etc. started producing the ".22 Extra Long" for rifles and revolvers. This cartridge used a 40 grain (3 gm.) bullet and a longer case that contained 6 grains of black powder in it. Then, the Stevens Arms and Tool Company combined the case of the .22 Long with the 40 grain bullet of the .22 Extra Long to produce the ".22 Long Rifle" (a.k.a. ".22 LR" cartridge).
Public domain image of different .22 caliber cartridges.
The .22 LR became very successful and still remains the most popular cartridge in the present day. Therefore when someone talks about ".22 caliber" in today's world, they are almost certainly referring to the .22 LR cartridge. However, back between 1857 and 1871, when someone said ".22 caliber" they would have been referring to what we now call ".22 short". So you can see how common cartridge designations have changed over the years.
For some early cartridges, they were named with two sets of numbers. For instance, right after the US Civil War, the US government issued the ".50-70 Government" cartridge for use with the Springfield Model 1866 rifle. This was later replaced by the ".45-70 Government" cartridge in 1873. So what do the numbers .50-70 and .45-70 mean? For a ".50-70", it means that the cartridge has a .50 caliber bullet and 70 grains of black powder. Similarly, a ".45-70" has a .45 caliber bullet and 70 grains of black powder. This naming scheme was also used for other cartridges such as the ".50-90 Sharps", ".50-100 Sharps", ".50-110 Winchester" etc.
In some situations, these cartridges were also referred to with three sets of numbers, such as ".50-70-450" and ".45-70-405" where the third number indicated the weight of the bullet in grains. This was done as cartridges of the same size, but with different bullet weights, became common. For instance, the .45-70 cartridge case was used for two different bullet weights and to tell them apart, the two cartridges were called .45-70-405 and .45-70-500, when the 500-grain-bullet variant was invented in 1884.
In the next section, we will look more into the conventions used to name cartridges.