I made my own styrofoam hot wire cutter using just scrapped stuff. I haven't purchased anything specially for this build, as I had everything lying around my house, so I cannot give you any prices for the materials. The idea is that anyone could make their own styrofoam cutter, adapting the tutorial with whatever suitable items one can find around. I really like the idea of upcycling, and I've tried to find a purpose for some items I was no longer using (old cd-player adapter, a metal plate from an old computer video card, pieces of copper cable, etc).

Step 1: Materials and Tools

You will basically need to make a sturdy frame which will hold a tensed piece of metal wire. This wire needs to be connected to a power source.

tools:
ruler, pencil, rubber eraser for measuring and marking
variable speed power drill for drilling and for driving screws
jigsaw
pliers

materials needed for the frame:
one piece of particle board (you can also use MDF, plywood, metal, anything which has a flat surface and decent strength and durability). Mine was a discarded 80cmx25cm, 20mm thick piece that I had from another project. A furniture workshop may give you scrap pieces for free or for a very small price. The arm of the cutter needs to withstand some considerable tension, so look for a material strong enough for that.
screws. I needed two types of screws. Some shorter than the thickness of my particle board (1.5mm), and some twice the thickness of the board or longer (I had some 45mm ones).
glue
a spring to hold the cable tensioned. Bike brakes have a spring that's very good for our job. You can use a regular coil spring if it's strong enough.
metal corner and flat plates. Computer cases have some slots on the rear panel which are covered with metal plates. They get to be replaced as you install various boards there. You can either use one of these cover metal plates, or detach one from an old video card, modem, lan card, etc.

cutting wire
I've used a nichrome wire that I've streched from it's coil. I've run it a few times against a piece of wood to straighten it. These wires are easy to get where I live from shops that sell household items. Lots of people here build electric heaters with them. You can alternatively use a guitar string, though I think it won't be just as good. Nicrome has a higher resistance so it will get hotter. It also has a higher melting point than steel.
Do not try to use copper or aluminum wire, as they have very low resistance and they will cause your other conductors to get hot as well, and you will also damage your power source!

power source
Although you could make a foam cutter running with alternative current directly from your wall socket, it would be extremely dangerous to work with and you may burn your house fuses. What you'll need is an AC/DC adapter capable of outputting at least 1 Ampere (1000mA), 2 or more Amps being preferable. I've found out that voltage doesn't matter that much. Phone chargers are usually not strong enough for this purpose. You'll need to read the specs from your adapter. A laptop charger should be good, but you may need to use a longer piece of wire or add some resistors on the circuit, because it may cause the wire to overheat. You may also not want to risk burning the charger of your running laptop. Some people have managed to adapt an old ATX power source to use for their styrofoam cutters. I've tried that, but the source got burned. You can also use batteries, but I find that a bit wasteful.
What I'm using right now is an 1A 4.5V adapter from an old cd-player. I've measured the intensity of the electrical current when my foam cutter was running, and it was somewhere between 1.5-2A, depending on how much wire length I was using. That means my adapter is used over it's designed capabilities and it may burn out unexpectedly from overheating. I try to switch it off whenever I'm not using the cutter, and so far it's running with no problems. It used to heat up much more when I was charging my cd-player, so I guess it will hold for the job.
The safest thing would be to use a variable power supply (you can buy one online for about 25$). A variable power supply would offer the advantage of being able to adjust the current, and thus the temperature of the wire. They also have fuses and protection circuits that would make your foam cutter safer to use.

electrical cables for connections
You need some thick wires to connect the power source to your cutting hot wire. If you use very thin wires, they will get hot as well and may burn out.

Step 2: Cut the Frame Parts

Mark and cut the board into four pieces. One would be the base plate, one the arm, one would serve to attach the arm to the base. One narrow strip would be cut into smaller bits serving as feet under the base plate. Try to make your cuts as straight and clean as possible, though some parts don't need a lot of accuracy.

Step 3: Attach the Feet and the Arm to the Base Plate

I've glued the feet on the base plate. I've attached the narrow piece of board to the base plate using screws. I've made some holes with a counter sink before driving the screws, so that the heads would not stick out.

I've marked the position of the arm. Make sure it stays vertical and that it rests well on the ground. When driving screws into the edge of the board, you should always pre-drill a hole so you won't have the board splitting up. Mine split towards the corner so I've patched it up with another nut and bolt screw which holds it together. It's probably good to use it anyway, because particle board is quite brittle at the edges. I've added another screw diagonally on the opposite side (pre-drilling and using the counter sink too).

Step 4: Attach the Metal Plate at the Tip of the Arm

Use short screws and have some pre-drilled holes for them. Make sure the tip of the plate sticks out a bit from the end of the arm.

Step 5: Bore a Hole Into the Base Plate

Use a ruler to establish the point where the wire will have to go through the base plate. Make sure the ruler is vertical. I bored a 4mm hole.

Step 6: Attach the Spring and the Underparts for the Wire

Turn the plate upside down. and put a corner plate above the hole. The wire will be ran through the hole on the corner plate, so try to position the plate so that the wire would come at the center of the hole.

Mark the position of the holes for the metal plates. I used a flat metal plate under the spring to avoid having the spring scratching against the surface, but the spring can be fixed directly on the board.

The string is fixed on the board with screws, but it should be easily taken out, so don't tighten the screws too much. In the end the spring will stay into place just because of the tension in the wire.

Step 7: Tie the Hot Wire

First tie the upper part of the wire. Run the wire through the base plate and the metal corner plate and tie it to the spring, leaving just enough length on the wire so that the spring will stay tensed at all times. You can then put the spring back into it's place. It should be well bent under the tension of the wire.

I've used some pliers to tie the wire. It's good to have some spare wire prepared in case the wire brakes when doing the knot.

Step 8: Time to Close the Electric Circuit

I had an universal adapter kit for my charger. I've chosen one tip that could be fastened to my frame with one small screw inside it's tip. This way, I could easily take out my adapter, when storing my foam cutter. It also allows me to easily change to another power source, in case I find a better alternative.

I think this part can be improved a little with a more elegant solution. The cables could be connected to banana plugs for instance.

A simple lazy man's alternative would be to split the cable coming from the charger and just tie the two cables to the ends of the nichrome wire.

Step 9: Make Sure the Connections Are Tight and Test

It's good to leave some extra length to the upper cable, so it can slide up and down the nichrome wire. This varies the resistance of the cutting wire, regulating the temperature. The shorter the wire, the higher the intensity of the current, and therefore the heat. If you need to get the wire really short in order to melt the foam, that means your power source is not powerful enough, and you run the risk of burning it because a wire which is too short makes the current really high.

If you're an electrical geek, you can add a potentiometer on the circuit to regulate the current. If you use a variable power supply, then everything is even more simple.

You can now start to cut your pieces of foam and experiment with the temperature.

Step 10: Safety Concerns

You shouldn't leave the styrofoam cutter powered when you're not around.

If you're using less than 2 amps or 12 volts, you won't feel any electrical current if you touch the wire, but watch out as it can get really hot. The metal parts which are out of the circuit (the metal plate on the arm and the spring) will not be traversed by electricity, so they will not heat up.

You should not leave the styrofoam cutter at the reach of children.

Styrofoam cutting produces toxic gases. They can give you head aches and, when exposed to them on a regular basis, they can give you cancer. It's best to use the cutter outside, or near an open window. If you cut foam inside, the smell will be persistent for hours, so it's best not to cut foam in your bedroom/living room. Wearing a mask is also highly recommended.