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There are an amazing variety of ammunition types out there and always more being created. There are some nuances so I expect someone more knowledgeable to jump in here and correct me if I get something a little off.

The most common ones are pretty straight forward. The main number is the diameter of the bullet itself. In the case of .44 that means the bullet is 0.44 inches in diameter as seen looking down the barrel. Likewise .22 caliber is about half the diameter. As basic geometry dictates it has much less mass. Beyond that there is a wide playing field out there about just how much powder is behind the bullet and how big the brass holding it is. Within a specific size there are variety. These are come up with originally by a gun designer and then adopted by other companies. For example .45 ACP stands for Auto Colt Pistol and was originally developed by colt. Just about every .45 automatic in existence uses this round so it is enough for someone to say .45 auto and people know what they mean. Another .45 is the .45 Long Colt. The diameter is the same but the shell is almost twice as long and is mainly used in revolvers. The rifle you shot was a 44 magnum. The real name for it is .44 winmag for 'Winchester Magnum'. Magnum generally means the cartridge is pretty long and differentiates it from an shorter wimpier round. In the case of the .44 that is just about the only flavor of 0.44 caliber bullet in use. There are others but just about any time someone says 44 it is know which one they meant simply because other forms or .44 are fairly rare. .357 is one where both a 'short' version and a 'magnum' are in common use. The shells are mostly the same but the 'magnum' is longer and holds more powder. Gun types toss around the numbers pretty free and easy mainly because there are only a relative handful of the most common types of bullet. It is mainly a matter of just getting to know them by what guns generally fire them. Another twist on it all is the metric bullets like 9mm. It is just the same thing only in mm instead of inches.
The number by it self does not tell you much except that people recognize that number and know what guns fire it.

.223 and .22 are about the same diameter but have vastly different power. G'nads rifle (the one 2j is posing with) is a .223. It is mainly recognition that allows numbers to be tossed around. We all know .22 and .223 but if you said .221 you would probably get a blank look and a question. Maybe someone somewhere in a lab made a 0.221 bullet but no one has ever heard of it. Like wise we toss around .44 and .45 and people will assume .44 winmag and .45 ACP are what you mean. Ask for a .43 and they will look at you kinda funny since no one has ever heard of such a gun. So while a .45 is higher than .44 since it is less powerful since by using those numbers I PROBABLY mean .45 ACP and .44 winmag. There is such a thing as a 45 magnum it is not at all common so that is most likely not what I am talking about.