What caliber guns does NATO use?

I presume you're referring to the 7.62x51 NATO cartridge, the civilian version of which is 308 Winchester. Both use a .308 caliber bullet of various weights.

Another standard NATO cartridge is 5.56 NATO. The civilian version of that cartridge is 223 Remington.

Both military cartridges have characteristics (bullet weight, powder charges, etc.) that may preclude complete interchangeability with some civilian/hunting firearms.

A quick note, in regards to the .308 Winchester being referred to as a civilian version of the 7.62x51 and the .223 Remington being a civilian version of the 5.56x45.. this is not true. The cartridges are dimensionally similar, but are not one in the same, varying in case thickness, neck angles, and other features.