What is background check for the graduate admission processing?
I don't know if this is universal, but I do know that some programs require background checks for anyone interacting with individuals under 18. For example, from the University of Pittsburgh Department of Psychology webpage (bottom paragraph on that page):


  Special note for Applicants to the Clinical Psychology Program: Consistent with Pennsylvania law, all students in the Clinical Psychology Program must satisfactorily pass required background checks that are intended to protect children from contact with individuals who have abused children or have been convicted of certain felonies (including drug felonies), as is further explained here: www.dhs.state.pa.us. FBI, Pennsylvania State Police, and Pennsylvania Child Abuse clearances must be completed during the fall term of the student’s first year. Students may not continue in the Clinical Psychology Program if they fail any of these background checks or decline to submit applications to obtain them. Cost of the clearances is to be covered by the student.


Given that many undergraduate students are underage, and given that as a graduate student you will almost certainly be interacting with undergrads as part of your teaching assistant and undergraduate mentorship duties, this is required for basically all grad students. I know that, at Pitt, this isn't just for the Psych program; lots of programs have a similar requirement.

Much of this is new since the Sandusky child abuse events.