That is a Siemens panel. They make a breaker that will replace
EDITone of those double pole 240v breakers with one unit that has four of the little half width breakers riveted together, the handles on the center two are tied together, and the handles on the outer two are tied together with a horseshoe shaped clip. Its two - double pole - 240v breakers in the size of one normal one. They are called "Quadplex plug in circuit breakers".
This will gain the space you need, Pull two of the existing 240V double pole breakers and replace with one of the quadplex, then install a standard double pole breaker of the size you need for the welder.
They are the bottom item on page 6 of the following PDF file.
http://www.sea.siemens.com/resident/docs/RPBR-S0007-1003.pdf
Below is a list of the ratings of the Inboard/Outboard breakers and the part numbers.
Inside two pole/outside two pole/part number
15A/15A/Q21515CT2
15A/20A/Q22020CT2 ***I think this is a misprint in the catalog, should be Q21520CT2***
20A/30A/Q22030CT2
20A/40A/Q22040CT2
30A/30A/Q23030CT2
30A/40A/Q23040CT2
40A/40A/Q24040CT2
50A/30A/Q25030CT2
Charles
Edit, didn't realize that the welder you intend to use is a 120V. In that case, on the same page of the catalog is a section of Triplex breakers. Its one 240v double pole handle tied in the middle, and two 120V single pole breakers in each outer position. From these two you can come up with whatever combo you want to replace what you have.
Look on that placard on the inside of the door, it will list all of the approved breakers for that panel.