It was essentially a biotechnology which acted as a bridge between a given Zygon and a life-form they wished to emulate, and imprinted the original onto the copy. The "originals" would be kept in stasis, attached to a machine. A Zygon could then access the machine remotely and request the shape of any life-forms then hooked up to it.
The individual whose body-print was being used had to kept alive as the likeness wore off every two hours, necessitating its re-assumption. (TV: Terror of the Zygons, PROSE: Sting of the Zygons, COMIC: Rest and Re-Creation) When killed, a Zygon would lose its link to the body-print machine and thus revert to Zygon form. (COMIC: Skywatch-7)
If the body-print machine itself were damaged or destroyed, Zygons could shapeshift, but only in a limited fashion. Using solely their natural abilities, they could maintain a given form for up to two decades, but they lost the ability to transform into anything else, or to revert back to Zygon form. (AUDIO: The Zygon Who Fell to Earth, HOMEVID: Zygon: When Being You Just Isn't Enough)