A Faraday Cage is a constructional concept named after Michael Faraday, the famous electrician from which we also get the capacitance measurement "Farad". The most basic an common use of such a cage is to block electromagnetic wavelengths from passing through a structure.
In design, a Faraday Cage is usually some sort of enclosure constructed of a grid of metal wires, traditionally in the form of a wire mesh. These need not be rigid or of any specific metal (though the type used will affect the cage's properties), only requiring conductive properties. They will be usually made to cover a small area, but there is nothing other than construction issues that dictate size.
A Faraday cage blocks transmissions based on the width of the lattice of the wire mesh and the wavelength of the electromagnetic waves in questsion. In electromagnetic field theory electromagnetic waves do not penetrate very far through holes that have less width then the waves wavelength. In corollary, the reflection of electromagnetic waves from a conducting surface is not much affected by holes (or other irregularities) in the surface with a size less than about a wavelength across.
What you get from this, then, is a surface that reflects electromagnetic waves that is not required to be a solid piece of material. Any waves projected from outside the cage will be either reflected off or diffused throughout the cage, and any created inside will bounce around inside until they decay or diffuse in a similar manner. However, the effectiveness of a cage will decrease as the wavelength size decreases to the point where it is shorter then the width of the holes in the surface.
Faraday cage design is also used a lot in Satellite Dishes -- Take the giant radio telescope at Arecibo; with the exception of walking and work surfaces, the dish is made almost completely of mesh plates. This allows the dish to be made much lighter without sacrificing much of it's receiving capabilities; while it cannot receive wavelengths less then about 1cm, those short of waves don't travel very well through intersteller space well anyway.
A Faraday cage need not be complicated, though they can be. It can be as simple as a metal pail (which is what Faraday first used in his research) or as complex as the stressed skin of a commercial jetliner.
Creating a practical demonstration of a Faraday cage is possible on a modest budget of money and time: all you need is a cage measuring about 1 meter square and 2 meters tall. Build the "walls" from 1cm pitch steel mesh, then secure the mesh a strong framework, except for the door section which would be best if made to hinge on part of the frame, and have a away to secure it closed. Place a (un)willing subject inside the cage carrying with them a fluorescent tube and secure the cage closed. Once they have calmed down and you have had your fun poking them with sticks while playing "capture King Kong" and whatnot, discharge a Tesla coil at them (you do have a Tesla coil, don't you?). Under normal circumstances the fluorescent tube would light, but since the cage is reflecting and diffusing the energy, the tube does not light (though no guarantees are made as to the subject wetting themselves).
Faraday cages are often embedded into existing structures where stray fields would cause problems, or where radio transmission is not desirable. Hospitals may use them in rooms containing electromagnetically sensitive instruments like MRI and PET scanners and of course the military and intelligence communities love to put them around rooms where work on sensitive information is done (because they block outgoing transmissions they are an effective way to stop bugging). Normal Humans also find them useful for everyday use (like putting wire mesh on a computer monitor to block the convergence-ruining field of a nearby electrical panel or whatnot).
Sometimes, a faraday cage can occur on its own in other structures -- a perfect example (that unperson reminded me of) is that an automobile shell is a Faraday cage that will protect the occupant in a lightning storm, rather than rubber tires.