Question about leaving supervisor
This is a fairly common situation, but there are lots of ways in which it can be handled.  A lot depends on the specifics of your situation and how well your advisor and your current department are getting along.  

One important issue is the source of your funding.  If you're currently funded by your advsior's research grant, and that grant is moving with your advisor to the new institution, then it is may not be possible for you to continue being funded through that grant as a research assistant at your current university but it might be possible to continue with that funding at your advisor's new institution.  

On the other hand, if you're currently funded by institutional or departmental sources such as a departmental teaching assistantship, you might be able to keep that funding where you are but it would not be possible to take it to your advisor's new institution.  However, it might be possible for you to obtain some kind of institutional or departmental funding at your advisor's new institution.  

There are three common ways in which this handled:


The student continues at the advisor's old institution and the old institution allows the advisor to continue to supervise the student as an adjunct faculty member of the old institution.  The student ultimately completes the degree at the old institution.  This works best if the student is close to completion and has funding that is independent of the advisor's grants.  
The student transfers to the advisor's new university and completes the degree at the new university.  This typically works well if the student is supported by the advisor's research grant and if the student is not very far along in the graduate program.  The new university would have to agree to accept the student into their graduate program.  The new university may have different course requirements and the student might have to repeat prelim exams or a candidacy exam.  Foreign students would have to get a new student visa.  
The student remains where they are and switches to a new advisor.  This works best for students who have departmental or institutional funding and aren't very far along in the graduate program.   


You should talk to your advisor about how he or she wants to handle this, and you should also talk to your department chair or the head of the graduate program to get the department's position. You'll also have to think about what you want- what's best for you may not be what your advisor or department wants.