The details are flexible. A hot wire cutter is simple. Take a wire and pump some electrons through it so that it heats up and bingo, it'll cut foam.
What we need:
A wire (the guitar string)
A frame (to hold the wire and put tension on it)
Moving electrons (the power supply)
Step 1: The Wire & Electrons
Lets talk about the power source:
Ideally you might have a variable DC power supply. In my case pumping 11.1 Volts @ 1.7 Amps did the trick. Varying the voltage will change the current. (V=I*R) The power (P=I*V) dictates the temperature of the wire. The more current, the more electrons are moving and smashing into the wire (resistance) and thereby heat it up.
If you don't have a power supply you should get one, they are very useful. But really, if you don't have one you can get buy with other power sources. Here are a few ideas:
1. A wall wart.
These are cheap and prevalent, but it will be hard to find one that can put out a couple amps. That is the problem with any power supply you try to use. The power supply that I have can put out 5A @ 18V which is pretty good. Usually, wall warts can do 0.5A which probably isn't going to cut it.
2. A hacked computer power supply
This is a simple hack. PC power supplies are all over the place. You can make a nice 5V/12V supply for next to nothing. Just search for instructables related to that.
Fixed vs. Variable.
Because we are using some random wire the voltage needed to drive the right ammount of current through the wire will vary. This is because the resistance of the wires will vary. So, ideally, we want a variable power source. This will also allow us to adjust the voltage and therefore the temperature of the wire if we'd like.
If your power source is fixed, that is no problem. We have another option. We can still adjust the current draw of the wire by changing the resistance of the system.