PYou have two big problems.
A squirreley ground
You can't put more than one ground wire in one hole in a ground bus(unless it's labeled to do so), and you probably can't do it with larger wires. The most important ground** on that entire bus (assuming it is a combo ground+neutral bus) is the bare wire that exits the panel and goes outside**. That is the grounding protection system. If it has problems, it will float your grounds, any ground fault will energize all your grounded places (chassis of machines etc.) compared to actual earth. A serious ground fault will float it a lot. And you've got a serious ground fault right now. So it really matters right now.
It sounds like somebody did some shoddy work and just stuffed the wires wherever, cranked down screws (if they did that) and hoped for the best. Nuh-uh. Fix all that and fix it right.
If you don't have enough room on your ground bus, you can buy additional ground buses from the panel manufacturer. Another (temporary) option is to pull several smaller grounds off, four at a time, and wire-nut them together with a pigtail, and put the pigtail on the ground bus. They even make ground wire-nuts that let you continue one wire through the nut. The pigtail should be the size of the largest wire, ideally do like-size wires together. Don't do this with neutrals.
A ground fault!
You are lucky that ground worked well enough to trip the breaker. You have a ground fault, or a "hot" wire coming in contact with ground. It may be in your wiring or your receptacle. Or it may be in the welder itself. Unplug the welder, and light up the circuit one more time. You better not see any sparks this time from the ground! If you do, you skipped the above step.
If it doesn't trip, you've now narrowed it down to the welder itself. I would first check its power cord and plug, and then off to the repair shop it goes.
** actually, as Tester101 mentions, the most important ground is the neutral-ground bond in the main panel. Since your ground bus is apparently overfull, and that's unlikely for only grounds, I assume this bus is being used for both neutrals and grounds. That makes the neutral-ground bond inherent, and so I am disregarding it.