Until now, I've only studied with diodes asuming they are ideal and I could tell the difference between normal diodes and Zener diodes. Now, I started to study non-ideal diodes and I see that this non-ideal diode has a breakdown voltage like the Zener diode. What are the differences between these two? Does this mean that the normal diode can work at reverse polarization like the Zener one?
The biggest practical difference is that a Zener* diode is made to break down in a useful and predictable way, while a rectifying diode is made to predictably not break down.
So if you buy a Zener diode that's designed to regulate to 12V, then you can figure that if you arrange for it to get the right current in the reverse direction, you'll get pretty close** to 12V out of the thing.
On the other hand, if you buy a rectifying diode that's sold to operate up to 100V, you can expect that over all the other advertised operating conditions of the diode (temperature, mostly), the thing won't break down at or below 100V -- but if it breaks down at 200V, it's still within specifications. You can also figure that if you design your circuit such that the diode does normally go into breakdown, you may not have a guarantee that the thing won't change it's characteristics over time***.
* "Zener" diodes work by a combination of the Zener effect and avalanche breakdown, mostly depending on the voltage they're designed to regulate. But in the market (in English at least) they're usually all called "Zener" diodes.
** generally \$\pm\$ 5% or 10%, depending on how much you paid for it, how close your current is to the design current of the thing, temperature, etc., etc.
*** A formerly-well-known hack for electronics hobbyists, back when we could all be counted on to have several dozen small-signal transistors in our collection, instead of a dozen microprocessor development kits of varying ages, is that if you connect base and collector of a small-signal transistor together, the base-emitter junction makes a pretty reliable Zener diode with a voltage of around 6-7V -- but I was always told that doing so damages the junction, and you shouldn't trust that transistor as a transistor thereafter.