I believe you can ship a handgun via U.S. Post Office - if you send it overnight delivery. That's all the ATF requires. All handguns must be shipped overnight and the receiver must have an FFL. That's my understanding.
Handguns can be mailed only by manufacturers and dealers and only between themselves. This is a matter of federal law. If a person wants to mail a handgun, that person can take it to his FFL dealer and have the FFL dealer mail it to another FFL dealer.
The law:
Section 1715. Firearms as nonmailable; regulations Pistols, revolvers, and other firearms capable of being concealed on the person are nonmailable and shall not be deposited in or carried by the mail or delivered by any officer or employee of the Postal Service. Such articles may be conveyed in the mail, under such regulations as the Postal Service shall prescribe, for use in connection with their official duty, to officers of the Army, Navy, Air Force, Coast Guard, Marine Corps, or Organized Reserve Corps; to officers of the National Guard or militia of a state, territory, commonwealth, possession, or district; to officers of the United States or of a state, territory, or district whose official duty is to serve warrants of arrest or commitments; to employees of the Postal Service; to officers and employees of enforcement agencies of the United States; and to watchmen engaged in guarding the property of the United States, a state, territory, commonwealth, possession, or district.
Such articles also may be conveyed in the mail to manufacturers of firearms or bona fide dealers therein in customary trade shipments, including such articles for repairs or replacement of parts, from one to the other, under such regulations as the Postal Service shall prescribe. Whoever knowingly deposits for mailing or delivery, or knowingly causes to be delivered by mail according to the direction thereon, or at any place to which it is directed to be delivered by the person to whom it is addressed, any pistol, revolver, or firearm declared nonmailable by this section, shall be fined under this title or imprisoned not more than two years, or both.
18 U.S.C. 1715
Congress gave the USPS authority to promulgate regulations regarding the shipping of handguns between manufacturers and bona fide dealers. The USPS did so with the Domestic Mail Manual.
http://pe.usps.gov/
The Domestic Mail Manual defines manufacturers and dealers as: "Licensed manufacturer and licensed dealer mean, respectively, a manufacturer of firearms or a bona fide dealer of firearms, duly licensed by the Bureau of Alcohol, Tobacco and Firearms of the Department of the Treasury, under the Gun Control Act of 1968 (Public Law 90-618), 18 USC 921, et seq."
Here is the language from the PS Form a person must sign to mail a handgun:
"The undersigned is a manufacturer of firearms or bona fide dealer therein, and the parcels presented for mailing herewith are customary trade shipments or other articles for repair or replacement of parts. To the best of my knowledge and belief, the addresses are manufacturers of firearms or bona fide dealers therein."