As Faheem pointed out there are other questions that deal with what a corresponding author is. You second question is not covered by those questions though.
The corresponding author should be an author that will be easy to find again after a few years. Ph.D students are not great choices since they are almost by definition not going to have the same contact information by the time most people read the article. In mathematics this is a serious concern because the backlog in articles can be multiple years and the student can already have graduated and moved across the world by the time the article is officially published.
As the other questions point out in some fields corresponding author is used to indicate contribution to some degree. But the old fashioned use of "the author you can find an address for" is still a worthwhile thing.