OR.....an FAA certification allows you to work "live" traffic on the individual positions you are certified on without an OJTI. (No licenses are issued in the FAA for radar controllers, tower only.) At the ICAO compliant facility I worked at you had an OJTI with you until you passed ALL your position checks and you actually had possession of youf license. Sometimes this could take a week or more. An FAA certified controller can technically continue right on position and work without an OJTI as soon as the supervisor giving the check puts an X in the box "certified" and signs the form. The ATCO receiving the check does not have to even sign the form.
It can mean other things too. I have seen several "certified" ATCO looking for their first ATC job. (Meaning they have completed their ATC school.)