There are many different "calibers" of ammunition out there. Strictly speaking caliber is the measurement of the diameter of the bullet out to the thousandth of an inch.
So, .308 caliber is nominally 0.308 inch diameter bullet
.323 caliber is 0.323 inch diameter bullet
Slang vernaculer for the above measurements varies considerably.
.308 caliber is commonly called 30 caliber and was principally a US caliber. Originating way back with military rifles in the late 1800's. There are many different "chamberings" based on 30 caliber or .308" bullets.
30-30 Winchester 300 H&H Magnum
30-40 Krag 308 Winchester
30-06 Springfield 300 Winchester magnum
Even though all the above chamberings fire the same .308 inch bullet, the brass case varies tremendously in size, shape, loading, pressure, etc. So a rifle chambered in 308 Winchester must be fed that EXACT type of ammunition, or very BAD things might happen.
To complicate matters the metric system is prevalent in Europe and many of their cartridges are labelled in metric measurement.
7.62 x 51 mm describes the .308 Win in millimeters rather than inches.
7.62mm = .308 inches
7.9mm = .323 inches
Slang vernacular 8mm milsurplus in general describes the 8mm mauser cartridge, which is nominally 7.9mm or .323 inches.
So, if you grabbed a 8mm mauser cartridge and put it into your 308 Win. chambered rifle and pulled the trigger.....
You are attempting to force a projectile of .323 inch diameter down a tube with a .308 inch opening. This will usually result in an explosion of your rifle, which may or may not injure/kill you.
Now whether an 8mm mauser cartridge will chamber in your 308 Win chambered rifle is another question. Sometimes the physical size of a cartridge prevents us from doing stupid things like putting the wrong round in a rifle but not always.
Hopefully I have not confused more than helped. Hopefully all the people who commented have impressed the importance of being very careful what terminology you use in describing cartridges. Using the term 8mm milsurplus is not helpful in knowing whether it will chamber and fire safely in your rifle. 8mm milsurplus likely means 8mm mauser but perhaps not always. 7.62x51 mm is just that it is not close enough to round off to 8mm<g>
I highly recommend getting and reading a reloading manual even if you never reload a round it is a wealth of info for the new shooter.
Shoot Safely, Mike