Its a punch from bare knuckle boxing where you you keep your jab hand above your head, and sort of unfold your arm towards your opponent, coming down at an angle. Its generally thrown after an outside step to put an angle on it, so you glance the nose or jaw. Its called an axe punch because you're chopping into your opponents face in an attempt to break the nose or stun them. It fell to the wayside with gloved boxing because its unlikely to "catch" the nose the same way with a glove as a set of knuckles, so I have no idea where Johnson picked it up. He keeps the hand high to throw the axe punch, and uses heavy glove defense with that hand, as its already near his face. The traditional axe punch is off the jab hand, but he also throws his right straight in the same chopping motion, probably to get over the glove.Glen Johnson Tribute
The whole thing is worth watching, but the best angle of the axe punch jab is at 2:55 against Roy Jones Jr and at 9:10 against Chad Dawson. Keep in mind hes throwing it at someone taller than himself both times, so the angle isn't what it should be, but you can see the chamber near the head and the downward motion of the punch.
Edit:Scottish Bareknuckle Boxing
I found this as well. This is different than the way it was used among the bareknuckle guys I've read about, but you get the idea. Here its a power punch thats more of a hammer fist or club with the forearm.