You could have the record expunged so that there is no public record of the arrest or charges - but if, in fact, you were arrested and/or charged for anything, a record of it would be the truth and not defamatory, even if the charges were later dismissed.
Often courts will order an expungement as part of dismissed/discharged diversion programs or deferred prosecution agreements, but if the court doesn't order it automatically, you can request it.
If you don't expunge the record, a background check could show DSMD (dismissed) or No Info Filed - Non-conviction (which means case was dropped) or something along those lines, and an employer would probably not be too concerned about it. But nothing should show up once your record is expunged.
Some government agencies will continue to have access to your record (law enforcement, for instance), and state professional licensing boards may have access to the fact that "something" was expunged from your record.
But, on the whole, expunged or sealed records will not show up on a routine background check and do not need to be reported to anyone anywhere.
I advise that everyone do a background check of their own on themselves to see exactly what shows up. It helps to know, in case there are errors that appear that can be corrected, or in case there are items that you may want to explain in advance on any application, so an employer/college/whatever knows you are not trying to hide anything.