A hot wire cutter is a tool used for cutting polystyrene foam. A hot wire cutter "cuts" by melting its way through the foam. It is intended for use with polystyrene foam and doesn't work with upholstery type sponge foam, polyurethane foam, hard plastic, etc.
The "hot wire" is Nichrome resistance wire. (ordinary thin wire will not work) The wire is held taut and has a low voltage passed through it to heat it up to just below red-heat. At this temperature you can slowly feed the material through and you will get a very smooth cut.
Never connect Nichrome wire directly to the mains. It must only be powered by a safe, low voltage power source such as a battery or plug pack.
You can buy various types ready made (example
), but building one is pretty easy.