This is the way I understood it.
My school did a background check on us. It was on the behalf of the hospital we did clinicals at. As long as we met the qualifications for the hospital we could do clinicals and therefore we could complete the nursing program. They didn't really tell us exactly what they were looking for, they just told us we had or hadn't passed the background check. Then, when I graduated and applied with the BON for my license I had to go to the police and be fingerprinted and submit the card to the BON for their background check which included an FBI background check. Then every place I have worked at do their own background check before I can start work. I would say everyone of those organizations have their own criteria as to what would disqualify someone.
My understanding of what would disqualify you were things like violent felonies like murder, rape, attempted murder, etc... would disqualify you for life. Also sexual offenses involving minors, or being on the sexual offenders registry. Most other felony charges such as drug charges, felony theft charges, things like that had to be older than a certain time period (I think it was 10 years). Misdemeanor thefts and crimes of moral turpitude and things of that sort had to be older than a shorter time period (I believe it was 5 years). There was something about DUI's, and I believe you could have had 1, but it had to be older than 5 years or something along those lines.
So, you might be OK on one background check, but may not be on another one. So, for example, you may pass your background check for your school, and even the board of nursing, but have a difficulty getting a job at certain locations as they may have different criteria. Like LaughingRN said, the BON won't care about what a school's requirements are, so they will not be of help to you as far as school goes. However, they should be able to tell you if you will be able to be licensed or not, which you will probably want to know before you go through school. I hope this helps.