When and why did boxing change from holding the backs of ones hands towards an opponent (the stance you see in the Notre Dame Fighting Irish logo) to the modern stance of holding the backs of ones hands towards a boxers left and right?

Boxing is, admittedly, outside of my area of specialty (I don't like getting punched in the head), but I can offer a bit of insight here.

Old boxers didn't really stand like the Fighting Irish guy. They stood more like this and this.

And here is a picture of two boxers squaring off: http://digital.library.villanova.edu/files/vudl:92985/LARGE

Notice how their hands are much lower than modern boxers. The forward hand is pointed toward the opponent, and the rear hand is curled slightly around the torso. The Fighting Irish drawing is something of a corruption of this.

The forehand is pointed toward the opponent because it would be used for a quick jab or a hook.

The rear hand was held in close to protect the body. It would also be used for close but powerful strikes. You see positions like this in many martial arts that are practiced without protective gloves. And gloves are what makes all the difference here.

Boxing gloves allow boxers to hit harder, and they allowed boxers to hit the head with dramatically more force. Boxers started holding their hands up to protect their head, because punches to the head started doing more damage than punches to the body (prior to gloves, body shots were more commonly used than they are today).

Holding the hands up, with the palms turn in toward the fighters face and with the knuckles against the temples protects the head from a heavy strike, while still giving some of the benefits of the forward hand for jabs and crosses.

Gloves also kept fighters from grappling, which was a common and important part of pugilism.

Anyway. I hope that helps a bit.

Sources:

Gladiators of the Prize Ring, by Billy Edwards.

The Complete Art of Boxing

http://hroarr.com/manuals/boxing-pugilism/Complete_Art_of_Boxing.pdf