If referring to a particular person we would say: what is the student's name. The apostrophe and the letter s is added after the word student, to show belonging. Basically, the name belongs to the student.
Generally this is the typical rule, even plurally, for example: what are the students' names?
However in some instances, there are exceptions. One example I am thinking of is if I were referring to a name of a category, for example, a class register might have a section called: Students Names. This is simply because it is acting as a title, rather than directly referring to anyone.
So essentially, when talking about a particular person or people, use the apostrophe with the letter s: student's names or students' names.
I hope this helps,
Shaun.