It means they will do a background check to see if you have a criminal record.
In this context, a bond is usually a written agreement by which a person insures he will pay a certain sum of money if he does not perform certain duties properly. In other situations, it might be an insurance policy used to ensure a legal representative will do his job and not misuse or steal funds he is controlling. The bond guarantees that a certain amount of money will be paid if a party is injured due to acts of the legal representative. There are many types of bonds for many different situations.
I property might require a general contractor to obtain a payment bond. If the owner pays the general contractor and the general fails to pay subcontrators, the subcontractors can make a claim agaist the bonding company.