Disclaimer: IANAL (I Am Not A Lawyer). I am a private pilot and have a current copy of the Federal Aviation Regulations (FARs).

This pertains to private pilots flying general aviation - NOT to commercial civil aviation. For commercial civil aviation you must comply with TSA rules, and whatever additional rules your airline might have.

There are two pertinent parts to the FARs: 91 and 135.

Part 135 pertains to commercial civil aviation. FAR 135.119 says nobody can carry a weapon on an airplane except for state officials authorized to carry arms, or those with the pilot's permission to carry arms. This is in addition to other regulations such as from TSA and the airline's own rules.

Part 91 pertains to general aviation (GA). If you are a private pilot, or you are flying with one (non commercial) then you are subject to FAR part 91, which is mute on the subject of firearms or weapons. This implies that if you are lawfully authorized to carry a gun (for example a CCW permit holder), then you can lawfully fly with your gun as long as it is not a commercial flight (which would be subject to FAR 135.119 and TSA regs).

But, to be safe, make sure of the following:
1. Have your permit in possession.
2. Do not enter the secure area of any airport (e.g. TSA areas).
3. Make sure you are legal to carry at both ends of your trip. That means wherever you take off, everything you fly over, and wherever you land.

Prudence would suggest having a lockbox in the airplane to secure your gun. This would be useful if flying over or into a place in which you are not authorized to carry. Having the gun unloaded and in the lockbox enables you to take advantage of the Safe Passage Act ( 18 USC 926A ) to legally travel through states that don't recognize the 2nd Amendment.

Of course, flying into or over Canada or Mexico would be a big no-no even with your gun in the lockbox. Canada is relatively accepting of rifles & shotguns (not handguns) as long as you pre-declare your firearms to customs. Mexico is not. And what happens in the case of an emergency, which might require you to cross the border, is anyone's guess. Since some IFR flight paths for US airports require entering Canadian airspace, I wonder what the legality is.

The bottom line: if you're licensed to carry, and you're flying GA (not subject to FAR 135), and your state doesn't list GA airports as off-limits, and you stay in states where you can legally carry, and you avoid the secure areas (if any) of the airports you visit, then you are legal. If you aren't licensed to carry, then you are legal if the gun is unloaded and in a lockbox ( 18 USC 926A ). This sounds like a lot of restrictions. But pragmatically speaking, it is not a problem for most private flights, since they tend to be local flights to and from GA airports.

ADDENDUM

Now you might ask, why would anyone want to, or feel the need to, be armed while flying? Why not always unload and case all weapons? GA flights do not have in-flight threats such as terrorists, violent passengers or UFO attacks. One is virtually never going to need the firearm while in flight.

One typically has a firearm to protect himself everywhere else - at his source, at his destination, if forced to land in a hostile area, etc. And if one already has the firearm loaded and holstered for this purpose, the question is, why remove it from the holster, unload and case it for the flight, then uncase, load up and reholster it upon arriving at the destination? All that becomes unnecessary handling of the firearm. And unnecessary handling is the cause of most accidental or negligent discharges. No firearm is ever going to fire itself while sitting in a properly designed holster. I am talking about modern pistols which are comprehensively drop tested by numerous federal and state agencies. By "properly designed" I mean a holster that entirely covers and protects the trigger area and has sufficient retention. It is safer to leave it holstered than it is to handle it unnecessarily. That is why one might decide to fly while armed - it is a practical risk management decision. It does not apply to long guns - rifles and shotguns. These should never be considered drop safe and should never be transported in a loaded condition. They should always be unloaded and cased.