In most cases, there are no federal laws requiring registration of a firearm. The exception to this rule is NFA weapons. NFA weapons include machine guns, certain parts of machine guns, short-barreled rifles, short-barreled shotguns, silencers, and destructive devices such as grenades or mortars. An additional category called "Any Other Weapon" is also included. This is a generic term used to describe a concealable weapon that can shoot, but doesn't quite fit into any other category. An example of an "any other weapon" would be a cane gun or pen gun.
To register an NFA weapon, you must submit an application to the Bureau of Alcohol, Tobacco, Firearms, and Explosives; a fingerprint card; a photo of yourself; and the applicable fee for your type of weapon.
Beyond NFA weapons, anytime you purchase a gun from a Federal Firearms Licensed dealer, you are required to undergo a federal background check and fill out a Firearms Transaction Record (a.k.a, Form 4473). While this is not technically the same as registering a gun, it is a record of the purchase.
As a collector, though, you may be purchasing many of your guns from a private individual. Except in the case of NFA weapons, no application process is required when buying a gun from an individual within your own state.
If you are buying a gun from out of state, however, you are required to let an FFL handle the transfer. In that case, the regular FFL process applies.
Although there are technically no federal laws requiring the registration of most guns, a few states require the registration of some or all firearms.