The main difference between an ordinary diode and zener is their REVERSE breakdown voltage.
Deliberately, the ordinary diode breakdown is made relatively high. And in normal operation, this breakdown voltage shouldn’t be reached otherwise some unexpected and/or undesired things may happen.
On the other side, zener diodes are made, deliberately, to have a rather known breakdown voltage (referred as zener voltage). In normal operation, the circuit takes advantage of it so that the voltage over the zener stays 'rather' constant independent of its current. But to use a zener properly, its maximum reverse current (Iz) that is allowed to flow in the zener diode should obey the formula:
Iz (in mA) < P_zener (in mW) / V_zener
where:
P_zener (in mW), power dissipation, is usually given on the zener datasheet.