Well according to the brochure here
http://www.millerwelds.com/pdf/spec_sheets/AD1-2.pdf it needs 52 amps for full power on a 230 volt circuit.
So you need a breaker bigger than 52 amps. The next size up is 60 amps. So you get a 230 volt breaker, rated 60 amps. (Note: when folks say 220 or 230 or 240 volts, its realistically the same here in the US, that's nominal voltage)
You could put it on a 50 amp circuit breaker, or a 40, or a 30...but if you try to run at full rated welding power, the breaker would trip.
Now that you've sized the breaker at 60 amps, you need to size the wire. The insulation class and how the wire is run (in a conduit or raceway) will determine the minimum gage needed to meet the ampacity. #6 NM is rated for 55 amps, but the Code would (in the past, I think it still does) let it be used with the next available size breaker (60 amps). Also the Code has a provision for welding applications for duty cycle, so in that case #6 would likely be sufficient.
If your cable run is very long, however, you will also have to factor in voltage drop. Say you're running a couple hundred feet from the breaker out to an outbuilding. The wire has an internal resistance, which means the voltage at the end will be less than the voltage at the breaker. If this voltage drop is excessive, you must go to a heavier wire. There are Java script calculators floating around the Internet which can do these calculations for you, enter the current and distance and wire size, they calculate the voltage drop.
Let's say the #6 winds up being OK. You have two #6 conductors and the smaller ground. The two conductors are wired directly to the breaker. The smaller ground goes to the ground buss. The ground normally does not carry much if any current, that's why it can be smaller. The predominant share of the current is carried by the two conductors that are tied to the breaker--this is a 220 (230, or 240, whatever you chose to call it) breaker.
If you aren't sure about this, don't feel like anybody's picking on you, but if it doesn't make sense, I would advise that you call an electrician.