I have read that zener diodes at some certain voltage let current flow through itself when it's reverse biased, but why can't the same happen in normal diodes? What is the use of zener diode?
The main difference is simply that a Zener diode has a specific, well-defined reverse breakdown voltage that's specified in the datasheet. Standard diodes don't tell you what their reverse breakdown voltage is, they just tell you it's guaranteed to be more than some set value--but a 100 V rated diode might actually break down at 100 V, or at 150 V, or at 4000 V. You can't know for sure, and it will be different even between parts of the same part number.
So you use a zener diode whenever you need a diode that breaks down at a known voltage, and a regular diode whenever you just need a diode.
There are several potential uses for this well defined breakdown voltage. The most common is as a voltage reference; you can use a zener and a resistor to get a stable reference voltage. This is how some linear regulators work, by comparing the output voltage to a zener diode reference voltage and adjusting the output to match the diode's breakdown voltage. Another application is setting a minimum voltage required for current to flow; I've used zener diodes before to make it so that a solid-state relay doesn't turn on until its input reaches 10 volts. They're also good for overvoltage protection, by shunting excess current away from sensitive electronics.
A functional explanation
Constant resistance. When a current I flows through a resistor R, a voltage drop V = I.R appears across it according to Ohm's law. This voltage drop is proportional to the current since the resistance is constant. So, if the current increases, the voltage proportionally increases and, if the current decreases, the voltage proportionally decreases.
Dynamic resistance. In the vertical part of the IV curve, all sort of diodes (ordinary, LED, Zener, etc.) behave as dynamic resistors keeping up a constant voltage drop across themselves. They do it by a simple trick:
If the current through the diode increases, the diode decreases its static resistance. As a result, their product - the voltage drop V = Iinc.Rdec, does not change.
If the current through the diode decreases, the diode increases its static resistance. As a result, their product - the voltage drop V = Idec.Rinc, does not change again.
With a similar trick, transistors keep up a constant current: if the voltage across them increases, they increase their static resistance and v.v. As a result, their ratio - the current I = Vinc/Rinc, does not change.