"Gadget" is a fairly informal word, and it usually refers to something small or hand-held.
"Gizmo" is a very slangy word. It suggests that you don't know exactly what the thing is: "How did he fix the heater?" "He had some weird gizmo with him."
I've only ever heard the word "appliance" used to refer to machines in kitchens -- refrigerators, dishwashers, microwave ovens, etc.