ok, so AC 101.
Coming into your service box there are three conductors.
2 hots and one neutral.
AC reads like a wave on an oscilloscope. So your voltage is the difference between the peaks and valleys. When you take a load to neutral that voltage is 110V.
When you combine the two loads you get 220V because the waves are opposite each other and now your distance between peaks is doubled.
and that's how single phase AC works in the simplest of terms.
I'd put a 50A plug on that circuit and rock out. How you wire it depends on the plug, but in general the hots are opposite sides and the neutral and ground are opposite top and bottom.
something like a 14-50 is what you're looking for at the hardware store.
that's 4 poles 50A 220V. Technically that's all kinds of wrong because your breaker is oversized. Whatever. Unless you want to change the plug on your welder to the 60A plug and buy a 60A receptacle. That would make sean happy.
So that gives you a hot on each side, a ground in the top (image is upside down, ground to the top these days) and the neutral in the bottom. Ground is the bare copper/green, White is the neutral. Black wire goes to the brass terminal. Red goes opposite on the silver one. Done.
Check with a meter between the hots for 220 and from each hot to neutral for 110 before you plug in your welder.
edit:
here's a better picture
