Homemade welders come in a variety of forms, but are often of one arc type or another. Microwave oven transformers are frequently used as a power source. The sheer quantity of examples available for building research online is nearly limitless, but here are a few for inspiration and to get you started in the right direction:

Homemade Stick Welder - From Microwave Parts! by Grant Thompson - "The King of Random"

Homemade Tig Welder by weldingtipsandtricks

Homemade Arc Welder 120v AC by kurtscottage

homemade battery tab spot welder by hsurfah

homemade spot welder by teslacoolguy

My Homemade Welder by junkman6261