A clean driving record is one that is free of accidents, moving violations or points. An insurance company also likes a record that is free of claims.
What an insurance company, employer, etc, considers a bad driving record can vary. A simple speeding ticket for only a few miles over the limit may not affect your insurance or be a problem for some employers however others may want a driving record completely free of any type of moving violation or accident, no matter how minor.
You would need to speak to the hiring manager of the job requiring the clean driving record to see what exactly the company determines "clean" to mean.
If your insurance rates have gone up due to a minor traffic accident you should comparison shop for car insurance here with us.