DISCLAIMER: If you already know all this, disregard.
I guess a good place to start is to recognize that the term "caliber" refers specifically to the diameter of the bullet which a firearm can fire. European rifles typically measure bullets in metrics - using millimeters; American rifles usually measure them in inches (the English system). Next, because most (not all) militaries use NATO-standardized ammo, their ammo will be often described in millimeters.
Now, having said that, it becomes important to realize that people often interchange metric and English terms for ammo, even if the bullets are the same actual size. A couple of examples:
-a .223 is in inches and a 5.56 is in millimeters, but the bullets are the same size. The "military" style rifles they are most often fired from are AR-15s, M-4s, etc.
-a .308 is in inches and a 7.62 is in millimeters, but the bullets are the same size. The "military" style rifles they are most often fired from are are M1A's, FAL's, and AK-47s. Follow?
Okay, part 2, briefly. Even though people talk about "caliber" when they say .30-06, .223, etc., what they most often mean is "cartridge chambering." Here's a brief example. A .30-06 hunting rifle and an FAL battle rifle both shoot a bullet that is .308 inches in diameter. However, the .30-06 has a longer brass case than the cartridge used in the FAL. So it's really important to know not just what diameter bullet is being fired, but what cartridge the rifle is chambered in.
Where it gets the most confusing is that ammo that was originally designated by NATO for military use has often made the transition to the civilian market, undergone subtle, but important changes, and should not be used in "military" (auto-loading) rifles. An example - is that a .308 winchester cartridge and a 7.62 NATO cartridge are VERY similar. Both shoot the same diameter bullet. Both have a case that appears identical (they're not identical, but look that way at first). The differences are mostly internal - the weight of the bullet, the amount and type of gunpowder in the case, and a few small but important dimensions to the case. Also, the civilian market of .308 winchester cartridges is very large, while there is only 1 real 7.62 NATO cartridge. You can fire a 7.62 NATO round from a bolt action rifle. However, because "military" style rifles are designed to shoot only the NATO round effectively, you should not fire a .308 winchester cartridge from a military rifle.
I write this only as a primer of sorts and an example, and I hope it's useful. You should start by choosing a cartridge based on what you will do with it. There is a ton of important information - just do some googling and read a lot. The details are what is important.