Gun permit laws vary by state. Some states require a permit just to own guns, others only require them to carry a concealed pistol. Some states allow pistols to be carried without any sort of permit at all! Depending on where you live and the kind of permit you need, you may have to take training classes, or simply submit a set of fingerprints and pay a fee.
When in doubt, check with a reputable local or state gun organization, or with local law enforcement. Typically permits are issued by your sheriff’s department or police depending on where you live. Some areas may even require special permits to own certain types of guns. A few states go as far as to require a special permit or license just to own a gun. You can learn more about gun permits and licenses here.