One answerer gave a good description above, but I will try to give you something just a little differently.
American ammunition is measured in calibers. 1 caliber is 1/100 of an inch. A .45 ACP is 45/100ths of an inch ( the ACP stands for Automatic Colt Pistol, which at the time was brand new and refers to a recessed rim that headspaces in the chamber).
European standards measure ammunition in millimeters. 7MM, 8MM,9MM etc. You can run a conversion and find out that the 9MM is .356 caliber, or exactly 1/100th of an inch small in diameter than the .357 magnum.
Now we get into some anomalies. A .38 revolver is NOT 38/100ths of an inch. It is actually .357 in diameter. It was classified as a 38 back at the beginning of the century and it stuck. A little bit later, law enforcement was wanting a more powerful shock ( stopping power) , so the .357 Magnum was invented. It is still the same diameter as the .38 (which is .357), but the case was made almost a 1/4 inch longer so it can hold more gun powder, therefor giving it more speed and power.
Now, let's look at another measurement that is commonly used. The AK47 fires a 7.62 X 39MM shell. This means the diameter of the bullet itself is 7.62 MM ( or 30 caliber) and the length of the shell casing is 39MM long. The .308 Nato round is 7.62X54MM. Exact same bullet but the case is longer by 15MM, therefor more powder. Now look at the 30-06 rifle round ( this was named due to it being a 30 caliber bullet and was invented in 1906). This round was the common rifle round used in American rifles until adoption of the Nato round in 1957. The 30-06 is 7.62X62MM. Same bullet as the two previous with the longer casing, again giving more room for more powder. But techically, you can stock the 30 caliber bullet in 147 grain ( the weight of the bullet itself) and use it to load three totally different guns.
One last item, since I probably have you more confused than when I started, is ballistic coefficients. This is the streamling of the bullet to cut the air as cleanly as possible. One of the best bullets in production today for long distance competitive shooting is the Sierra Palma Match in .308 caliber ( we are back to the 7.62MM again.). It has a B/C of .49. The lower the number the more streamlined the bullet is and it will shoot "flatter" than another buller with a higher B/C. Just for basics, the pointier the bullet, the cleaner it cuts the air. As the bullet nose gets blunter ( as in the 30-30 round), the worse it will cut the air and it then turns into what is called a "rainbow round", meaning that on a distance shot, you fire HIGH into the air and the bullet will travel in a huge arc to hit it's target. The 30-30 is an excellent example of a rainbow shooter. Look at the ballistics chart and you will see the weapon is sighted in dead at 100 yards. To hit a target just 350 yards away, you must fire at an imaginary point 28 inches above where you want it to hit. It is going to drop that far going an extra 250 yards.
This is all part and parcel in both reloading and learning to shoot distances ( part of sniper training). Beyond this is the basic types of powders ( extruded, ball and flake) which refers to the basic burn rates. Rifles use much slower powders and will burn as the round goes down the barrel. This is done so the chamber pressures won't go so high that it will explode the barrel. Short barrel pistols use flake powders, which burn the fastest. All the force is expended when the trigger is pulled and the bullet is away. You use standard primers with fast powders and magnum primers for slower rifle powders. The magnum primer is a much hotter primer and is needed to ignite the slower burning powder. If you use a magnum primer with a fast powder ( such as Winchester 231), you will develop dangerously high head space pressure and damage ( or blow up) your weapon.
I think I have preached enough here. If you want to know more, write back. I hope this helps and you can find it useful