The Faraday cage is a electrical phenomenon of conductors discovered by Michael Faraday in 1836. It is a property of an enclosed conductor that says that if the conductor is placed in an electric field, charges will be distributed along the outside of that conductor that will in effect create an opposing field within the conductor that cancels the external field leaving the inside of the conductor empty of any electrical field. As you can see from the animated picture below, first the enclosed conductor is introduced to an electric field. Charge separation then begins within the conductor. As charges accumulate on opposite sides of the conductor, a new field 180 degrees off the external field is generated within the conductor that cancels the external field leaving the potential within the cage at 0.
Now lets talk about electromagnetic radiation. If we construct a faraday cage from a mesh type metal material and stand inside with our cell phone, it will in fact block any transmissions into or out of the cage assuming the cage meets a certain requirement. The mesh material must have small enough holes to effectively block the wavelength of the radiation attempting to penetrate the cage. The cage itself is also grounded in order to dissipate any current created.