I am building a shed 150 feet from my house and would like to run my 220 volt Miller stick welder; what size of wire would I need to bury underground? Thank you in advance.

Hi,
I cannot read the tag on the machine, so about the best I can do is give you this link:
Electrical Wire Sizes - What Size Electrical Wire Do I Need
And say that you are going to experience 150' of voltage drop. You need a UF rated cable to bury it.
Peace,
Clarke