The late, great P.O. Ackley was one of the most prolific wildcatters of the 20th century. He made an Ackley Improved version of most of the current factory cartridge of the period plus a bunch of other cartridges as well.
His basic recipe for the common Ackley Improved version was to blow the case walls out so they're nearly straight and change the shoulder angle to an even 40°.
The measurement from the datum line on the neck to the head remains unchanged (or set back just slightly) so that the parent case is properly headspaced. This way you can still fire the parent cartridge in the Ackley chamber to fireform it...or if you run out of AI ammo while away from home on a hunting trip.
You get a little extra case capacity with the AI version, the exact reasoning behind it is always debated. Some say he did it to squeeze more velocity out of the case, but others say the changes are made for a more stable cartridge case (which is how I understand his writings).
Even if Ackley himself didn't develop it, someone doing a straighter-walled, 40°-shoulder improved version of a cartridge will almost always label it an Ackley Improved version because it follows his system.
Here are a couple of scans from the Sierra manual of the .257 Roberts and it's AI version. If you compare the dimensions you can see basically what's all been done.
Some time, you can rechamber to the AI version of a cartridge simply by recutting the chamber with the AI reamer, but most of the time you also need to set the barrel back a few thousands so that you get a clean cut and can set the proper headspace.
I know several guys who've gotten AI jobs done but the chamber was cut too long to properly headspace the parent case afterwards. Not a big deal if you're a handloader and you're aware of it when fireforming cases, but it can catch you off guard though too.
Even if you're not a wildcatter or reloader, Ackley's Handbook for Shooters & Reloaders (a 2-vol set) is still must-have reading material. If you ever find a copy, check it out.