Turns out the plug we had is NEMA 6-50. You can probably figure out what yours is from a chart. If it's close, measure the lengths of the prongs and compare to what is listed. Once you know your plug type, find a mating receptacle on Amazon or somewhere else - probably about $10.
Now go check your breaker box. In most US residential neighborhoods, you'll have two thick wires coming in from the street. These are the mains 240V alternating current (oscillates between 120V positive and 120V negative). Each of the 120V AC outlets in your house uses one of the lines from the 240V pair and one "neutral" line (which has a 0V potential), so the total potential at the outlet is 120V. But we want 240V so we'll have to draw from both lines (poles) at once. For that, we'll need a double throw (2 pole) 50 or 60 Amp (lots of current for a welder!) circuit breaker. Also available on Amazon for about 10 bucks.
Before fiddling around in the box, turn off the main breaker where the mains lines come in!
Now fit in your new breaker - it's big enough to span two of the tabs protruding from the rails in the back of the box (one from each pole).
Get three lengths (one of each red, black, and green) of AWG 8 insulated copper stranded wire (again, very thick for the big currents we're handling) from your Home Depot or wherever. Strip the ends a centimeter or so.
I only needed to run my wires a few feet into my garage, but you may have to route a longer distance. Poke out another one of those tabs in the side of your breaker box to make room for the new wires.
Screw in your green wire into the ground rail. Screw your red and black into your newly installed breaker. Screw the other ends of the wires into your receptacle. If you're not installing the receptacle in a wall, be sure to cover the back with some insulating tape to ensure that you don't accidentally short the two liver terminals.