Yes, they work. Just make sure that you have one that will read the satelites in your area. I'm in Germany, and not all US GPS systems will work here. No, there is no monthly fee.
Yes there is a fee as it is a very costly system to avail of.GPS chips are inserted into your phone and the satellites track down your cellphone.
The short answer is that you have an electronic gizzmo that finds/is found by at least 3 satellites. The distance between them is calculated and it can tell you where you are. There is no fee for a regular GPS unit. There are fancy ones that can be put in cars that may. The main thing that may be beyond the upfront cost is that some of them can have new maps put in, and that would be extra. The long answer (these guys are great): http://electronics.howstuffworks.com/gps.htm
There is no fee to use GPS. GPS consists of a 'galaxy' of satellites in geostationary orbit (thanks to the US Military) that transmit signals that are picked up by your GPS receiver. The software in your unit computes your position based upon which satellites your unit is receiving signals from and the information in those signals.
