Switching and Zener Diodes
Zener diodes are constructed in a way so that in reverse-bias the zener effect causes current to flow. This can be controlled (at time of manufacture) to a certain voltage. In forward bias the operation is identical.

Wikipedia page on Zener effect (tunneling)

Signal diodes also break down in reverse bias but that is due to avalanche breakdown which may appear similar but is caused by something completely different. It is also generally at a higher voltage.