Federal Law requires that all modern firearms be shipped to a holder of a valid Federal Firearms License (FFL) only. The recipient must have an FFL; however the sender is not required to have one. Any person who is legally allowed to own a firearm is legally allowed to ship it to an FFL holder for any legal purpose (including sale or resale).
Here is exactly what the ATF ‘Federal Firearms Regulations Reference Guide’ (ATF P 5300.4) says:
(B9) May a nonlicensee ship a firearm by carrier?
A nonlicensee may ship a firearm by carrier to a resident of his or her own state or to a licensee in any state. A common or contract carrier must be used to ship a handgun. In addition, Federal law requires that the carrier be notified that the shipment contains a firearm and prohibits common or contract carriers from requiring or causing any label to be placed on any package indicating that it contains a firearm.
[18 U.S.C. 922(a)(2)(A), 922(a) (3), 922(a)(5) and 922(e), 27 CFR 178.31, 27 CFR 478.31 and 478.30]
(B8) May a nonlicensee ship a firearm through the U. S. Postal Service?
A nonlicensee may mail a shotgun or rifle to a resident of his or her own state or to a licensee in any state. Handguns are not mailable. A common or contract carrier must be used to ship a handgun. A nonlicensee may not transfer any firearm to a nonlicensed resident of another state. The Postal Service recommends that longguns be sent by registered mail and that no marking of any kind which would indicate the nature of the contents be placed on the outside of any parcel containing firearms.
[18 U.S.C. 1715, 922(a)(3), 922(a)(5) and 922 (a)(2)(A)]
‘Antique’ firearms need not be shipped to a licensed dealer. These can be shipped directly to the buyer. An antique firearm is a firearm built in or before 1898, or a replica thereof. The exact ATF definition of an antique firearm is:
Antique firearm. (a) Any firearm (including any firearm with a matchlock, flintlock, percussion cap, or similar type of ignition system) manufactured in or before 1898; and (b) any replica of any firearm described in paragraph (a) of this definition if such replica (1) is not designed or redesigned for using rimfire or conventional centerfire fixed ammunition, or (2) uses rimfire or conventional centerfire fixed ammunition which is no longer manufactured in the United States and which is not readily available in the ordinary channels of commercial trade.
Knives, air guns, accessories, and most gun parts need not be shipped to an FFL holder. We say most gun parts because each firearm contains at least one part that the ATF considers a firearm. This part is typically the part that contains the serial number. This part must be treated as a complete firearm when shipping the item.