"Caliber" is the term for size designations for bullets (projectiles) and the inside diameters of the gun barrels through which the bullets are fired, as well as for the complete cartridges (rounds) and the area--called the firing chamber--in which the cartridge is inserted at the back end of the barrel preparatory to the bullet being fired out the barrel.
"Gauge" is a term for a similar purpose in relation to shot shells (for shotguns).
Caliber is expressed in units of either inches or millimeters (mm).
INCH DESIGNATIONS FOR CALIBER
A designation like .22, .25, .308, .32, .357, .38, .380, .40, .44, .45, or .50 is an expression of the approximate outside diameter, in inches, of the projectile (bullet) part of the cartridge and of the inside diameter of the barrel. The designation typically has some word(s) or abbreviation associated with it.
For example, ".22 short" refers to a .22 caliber cartridge with a short case and a light powder load. Other options of such .22 caliber "rimfire" cartridges include ".22 long, .22 long rifle (LR), and .22WMR ("magnum"). A .22 (caliber) rifle chambered for .22LR is commonly (but not always) able to shoot the short, rifle and long-rifle cartridges.
The case of the "magnum" is both longer than the others and has a diameter somewhat greater than the actual projectile diameter, so the magnum cartridges will not fit in a gun chambered for .22LR--and these cartridges produce considerably greater energy and projectile speeds.
The words "magnum" and "super" in caliber designations always mean more projectile energy than for cartridges without the word. The ".357 magnum" is a caliber for a cartridge that is closely the same as the ".38 special" except for the fact that its case is longer and contains more gun powder, and therefore produces greater projectile energy than the .38 special. The .38 special is the caliber for the revolver that was long the standard weapon for police and for civilian self protection. It and the .357 magnum are still much used in this manner.
A .357 magnum revolver can shoot .38 special cartridges in addition to the "mag" cartridges. This is done often for target practice because the smaller cartridges are cheaper and don't "kick" as hard. The .357 magnum is also used for hunting by people who like to give the animal more of a chance than they would get from a hunter with a rifle (the handgun hunter has to get closer, because the handgun is not as accurate as a rifle).
A cartridge with a very slightly smaller outside diameter than the .38 special is the .380 auto for automatic and semiautomatic handguns. The .380 cartridge case is shorter than that of the .38 special and holds less powder, so the cartridge does not produce as much energy as the .38 special.
Some of the inch calibers mentioned earlier generally refer only to cartridges for automatic or semiautomatic handguns. For example, the .25 and .380 are common calibers for such handguns but not for revolvers. Others calibers are common only for revolvers (like .38 special and .44 magnum). For some, however, there are cartridges for both types of guns. An example is ".45 Colt" for revolver and ".45 ACP" for automatic and semiautomatic. For some numerical designations there may be several different calibers differentiated by the appended word(s)/abbreviation.
".223" is a small caliber originally developed for warfare. ".270 Winchester" is a commonly used big-game hunting caliber. For some numerical designations, again, there are several different calibers that are differentiated from each other with word modifiers of several types.
There are also numerous rifle calibers that are identified with usually a two digit decimal number followed by a dash or a slash, then another number. Examples: .22-250 Remington, .30-.30 Winchester, .25/06 Remington and .30-06. The meanings of these second numbers are not consistent. For the "30-aught-six" the 06 refers to the year of invention, 1906.
METRIC DESIGNATIONS FOR CALIBER
Handgun calibers are commonly identified as something like "9mm Luger" or 10mm. These both are cartridges for automatic and semiautomatic handguns.
Numbers like "7.62x33mm" are metric designations for rifle calibers. This one is a caliber equivalent to the ".30 caliber carbine" cartridge. The 7.62 is the diameter of the projectile in millimeters. The "33mm" is the length of the cartridge case. Another example is "7x57mm Mauser."
SHOTGUN AND SHOT SHELL GAUGE DESIGNATIONS
One shotgun guage, the ".410," is a decimal inch designation for inside barrel diameter (i.e., the "bore"). Other guage designations are different. There are 10 guage, 12 guage, 16 guage and 20 guage. The larger the number, the smaller the barrel inside diameter. The numbers are the numbers of spherical lead balls--of diameters equal to the inside diameter of the gun--that are required to total up to a pound of lead. The greater the barrel inside diameter, the fewer balls are required to make the pound.
This should clear up a lot of confusion for most folks that know little about firearms.