Background check before employment offer What's the standard procedure in University jobs when it comes to "background checks"?
Do they call my current employer or is it limited to a "criminal background check"?

Edit: tenure track job in the US
I'd expect the following:


Some institutions do a criminal background check, others don't.  This will usually be disclosed in the job ad.  For those that do, they might do the check before making an offer (in which case the application will ask for the necessary identifying information), or they might make you an offer that's contingent on passing the check.
US faculty jobs will require letters of recommendation, or contact information for references.  The search committee might actually contact them to ask further questions about you.  (Your recommenders might or might not tell you if this happens.)
They will normally not contact anybody other than your letter writers or references you list in your application.  This is pretty important because people in existing faculty positions often don't want it known within their department that they are on the market; it could have political blowback if they end up not leaving.