For civilian firearms, they are usually marked on the gun. You will probably need to pay a gun shop a visit for some hands-on advice.
Mainly from the markings on the gun by the maker.
Contact the maker.
Find books on the maker.
call the maker.
by stalking him
Gunsmith, gun show, gun shop, pawn shop, or old catalogs of the maker
Find a book with sn's from that maker.
You will have to contact the maker to find out.
Most guns actually have their maker on the gun itself. Check the side of the gun.
Find out who the maker of the gun is and see if they can help you with the model name and number. If this is an old gun and the maker is no longer in business check with a professional appraiser. The number by itself does not mean much.
Post the maker's name and serial number.
List the maker, caliber, type and serial number.