The university where I'm doing my PhD is asking me to register a co-supervisor. However, I don't really know the role of co-supervisor. What exactly is it for? Does it mean that for every publications we do, we should also add him in the authorship?!
There are several reasons that the university wants you to do this.
The most serious is as a safeguard against possible problems that might arise if your relationship with your (actual) advisor deteriorates. For example, if your advisor asks you to do something that you think is unethical, or is making what you think are unreasonable demands on you, the idea is that you should be able to go to this second advisor to talk about these things. Whether or not this is actually realistic is another question.
This has nothing to do with authorship of publications, unless the second advisor is actually an author.
There are a few possible reasons.
One of the most important aspects of your graduate education will be how to successfully collaborate with others. By requiring you to work with someone other your boss, they hope to have you learn more about working on a research team.
It's almost always a good idea to have many eyes on research before it goes out, to make sure that it's inspected from a number of viewpoints. Requiring a larger committee enforces that idea.
Having someone else as a "dotted line boss" provides you with someone else who is officially responsible in some way for you. This can be a good way to generate discussion and get feedback on your ideas from someone other than your advisor. The most useful discussion I had in the entirety of my graduate career happened with one of my committee members, not my advisor.
Some programs require you to have a co-chair not in your program. In that instance, your graduate program would like to make sure you leave the program with exposure to disciplines other than your own, and this policy enforces that.