The idea behind all underground pet fences, regardless of the manufacturer, is the same. In all of these systems the transmitter uses the buried loop of wire to broadcast a radio signal. The signal is normally very simple -- just a sine wave, or possibly two sine waves at different powers. As described in How Electromagnets Work, the buried wire acts as an antenna and turns the signal into electromagnetic waves.
The transmitter does not use a lot of power, so the signal around the wire has a very small range -- perhaps 10 or 15 feet. In some systems the wire has two signals running through it -- one at low power and one at a higher power. In these systems the collar can detect different distances from the wire to provide different levels of correction.
Inside the collar is a small radio receiver (essentially an AM radio very similar to a $5 battery-operated AM radio you would buy at a discount store). When this radio gets close enough to the buried wire, it receives the signal that the wire transmits. The radio triggers a correction so the dog knows it is nearing the boundary.
The underground wiring installed around the perimeter of your yard loops back to the transmitter that you have plugged in a safe, dry location (it is also recommended that you install a lightening rod near the transmitter to prevent blow-ups). See the sample layout for an example of this looping. The transmitter controls determine how wide the electric field is from the installed wire.