Background checks show a couple of things. First they show that you were arrested or charged in some matter. Then it shows the disposition of the charge. If you were arrested and the matter was dismissed, the arrest remains on the record and is accessible to those performing a background check. You should keep the documentation of the dismissal handy.
I must agree with the first answer given. While criminal records should not show arrests, they often still do and that should be changed. However, I've been complaining about that for over 35 years and it still hasn't been changed, so don't expect it too soon. Like the first answer said, keep all your documentation available for the next job interview, You may be able to expunge such an arrest record, but its unlikely. Speak to a local lawyer if you wish.