I have an old Ariens snowblower and this past weekend the weld that holds the auger to the center axle broke.Not sure if that is clear on where the weld is... My question is what kind of weld will work to fix it. Not sure on the differences between the different types of welding.

Standard "stick" welding will work fine - provided surfaces are prepped properly.

I've done that repair on some units - the hard part is returning the auger rake to it's original position were is was welded originally. Getting the rake straight (curved the way it was) is not crucial, but if not "straightened" enough on some models will protrude farther out than original and contact auger housing, possibly preventing it from making the rounds.

When you say "stick" welding do you mean arc?