A "air gauage" is a precision sized slug with holes in the side. Compressed air is pushed into one end, and the flow rate is measured. The larger the flow the greater the side clearance. They are calibrated with precision ground "rings" IE: a .0005 clearance might flow 0.1 cubic inch/minute and a .005 clearance 1.0 cubic inch/min. This gives you a way to measure the variation in diameter, along with the diameter itself as the gauge is passed through the barrel.
Any barrel can be "air gauged", the differance between the "cheap" barrels and the "prememum" barrels is the amount of variation (tolerance) the manufacture allows before relagating the barrel to the scrap bin.

It's not really for measuring the straightness of a barrel, it's for measuring the consistency of the bore/groove diameter. The most popular custom barrel makers hold their best barrels to something like .0002" from end to end. Whether it's a little tight or a little big in bore/groove diameter isn't as much of a concern as it being consistent throughout it's length. Some even prefer a little "choke" on the muzzle end.

Good info here, thanks, Ken

WayneShaw has it right. Douglass "air gages" all of their barrels. Those that exhibit no more than .0001" variation, are labeled "air gaged". Also, they mark the end with the tighter bore "muzzle".