Brian
If the name on the gun is indeed AMERICAN GUN COMPANY and not just AMERICAN or AMERICAN ARMS CO or AMERICAN BARLOCK WONDER or AMERICAN BOY or AMERICAN CHAMPION or AMERICAN FINE DAMASCUS or AMERICAN GUN COMPANY (word spelled out) or AMERICAN GUN CO KNICKERBOCKER or AMERICAN GUN CO OF NEW YORK or AMERICAN GUN CO OF N.Y. or AMERICAN HAMMERLESS or AMERICAN NITRO or AMERICAN SIDE SNAP or AMERICAN STANDARD or AMERICAN STAR then it was made by the Crescent Fire Arms Company of Norwich,CT (1892 to 1930) for and was sold by the Supplee Biddle Hardware Company of Philadelphia, PA, a wholesale sporting goods dealer. It is called (by me anyway) a "Trade Brand Name" shotgun. A "Trade Brand Name" shotgun is an inexpensive shotgun that was made by a major maker (and before 1940) usually on contract for a wholesale sporting goods dealer, a retail chain store or an independent seller who chose the name to go on the gun. As said, these were inexpensive but serviceable shotguns. You didn't mention if the gun has outside hammers or not but if it has hammers it was made in 1903 and if it has no outside hammers, it was made in 1908. A word of caution at this point. These guns were designed and made using the technology and metallurgy of the times and were designed for the ammunition in use back then. While the gun may or may not have damascus barrels the ammunition in use in 1903 and 1908 was black powder and maybe, just maybe very early low pressure smokeless powder and lead shot loaded 2 1/2 inch shells. The guns were not designed for longer, 2 9/16 and 2 3/4 inch shells and certainly not 3 inch or magnum shells loaded with high pressure smokeless powder and steel shot. Since I can't see your gun to examine its condition I must recommend that you do not attempt to shoot it. Some people do shoot these old gun and if you are going to try to, then please have the gun checked out by a good competent gunsmith before you do and use appropriate ammunition. Value? Everyone wants to know. The value of any of these old guns depends on its condition, the amount of original finish remaining on the metal and wood as well as the mechanical condition. A prime condition example (rare) that appears to have come out of the factory yesterday afternoon might bring as much as $125 while a rusty metal, rotten or broken wood and missing parts piece of junk fit only for parts salvage or as a a canoe paddle might bring as little as $10. As you say, sentimental value is priceless.