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