Many employers do criminal background checks and do not employ persons with criminal records. There is no specific law against generally hiring people with a criminal record. A criminal record may disqualify a person from certain kinds of employment or from professional licensure.
There are several court decisions that have held that blanket prohibitions on hiring employees with criminal records may violate Title VII of the Civil Rights Act of 1964. You may want to discuss your situation with an attorney to determine if this would apply to your situation.