The cables are awful. The positive terminal has a rubber hood covering it, that thing is old and ratty. There seem to be two conductors leading from the positive terminal -- one goes to the alternator, the other goes, presumably, to the rest of the car's circuits. Both of these conductors have sheaths which look like... donkey butt. I'll get them replaced, but for the immediate future I just want the thing to work again.
You can usually get by with cleaning the cables, but it won't work if the conductor has been compromised by the corrosion, obviously. Worse yet, corrosion can spread up into the wires, and it would be a good idea to cut back the insulation and check for corrosion there, as well. If you do find it there you'll need to replace the wires, of course. Finally, after cleaning the terminal, or as you're replacing it, be sure to apply an anti-corrosive substance to both the battery terminal and connector before reattaching it.