Zener diodes are only a slight variation on ordinary diodes, so to understand zener diodes you must first understand how diodes work. Semiconductor diodes are made out of two materials: P-type semiconductor and N-type semiconductor. One wire at each end of the physical device is connected to each semiconductor type. The semiconductors are physically joined in the middle at the NP junction. Semiconductors are crystal-like structures--very pure except for a few carefully added atoms. If one type of atom is added (in a process called doping) you get P-type semiconductor, and if a different atom is added you get N-type. In N-type semiconductors, the atom that was used for doping has one more electron than the atoms in the crystal; therefore, there are some unattached electrons floating around. In P-type semiconductors, the atom that was used for doping had one fewer electron than the atoms in the crystal, leaving a hole where an electron belongs, which floats around. With forward biasing, the electrons and holes are pushed together and current flows. With negative biasing, the electrons and holes are pulled further apart and no current flows. Under reverse biasing (no current flowing) a lot of energy is going into the diode. If there is enough energy and it lasts long enough, the diode will break down.
Over-doping can cause the breakdown voltage to occur at precise voltage levels. These breakdowns occur long before the energy levels become harmful. Zener diodes have exactly this ability. A forward-biased Zener diode is exactly the same as a forward-biased ordinary diode--current flows with little resistance. Reverse biasing is different. Starting around zero, reversed biasing looks like an open circuit (very large resistance). Both Zener and non-Zener diodes strongly resist small currents of reverse biasings, but at some level (called the breakdown point) the Zener diode allows current to flow freely in the reverse direction. The behavior is normal for a Zener diode and the "breakdown" does not damage the diode.
Because the breakdown voltage can be precisely controlled by adjusting the doping of the semiconductors, Zener diodes are useful in places where a reference voltage is needed. These places include power supplies, voltage regulators and some amplifiers. Zener diodes can also be used as a "clamp" to insure that an AC circuit does not exceed safe voltage limits.