How to control appliances with arduino?

Different things...mosfet vs. bjt. Either can work, depending on the circuit and the part you select - and how you wire it of course.
It's not rocket science.

The difficult part usually is to establish with sufficient certainty how exactly the switch you want to replace/manipulate is wired to the rest of the control electronics. Sometimes it's straightforward and a visual inspection and a poke with a multimeter already gives solid answers. Sometimes it's not that simple. You never know until you start opening stuff up.