How can someone check a gun's background?

Some are easier to research than others. For example there is a Remington Collectors Society, but you cant find much information on Remington serial numbers. On the other hand if you have a Winchester serial number you can usually find when it was made without much difficulty. google for the make of gun you have + collectors. Some of the manufacturers will research the gun, for a fee, and send you a Factory Letter, stating when the gun was made, and where it was sent from the factory.