What is "Director" in ATC?
Some TMA arrival sectors can be further split up into an arrival/pickup part and a director/feeder part.
The arrival controller is responsible for getting aircraft separated either onto downwinds or onto transitions, where they are handed off to the director. The director is then only responsible for turning these aircraft onto the approach, in most cases the ILS.
Essentially, the arrival or pickup tries to get everyone onto the correct separation and speeds, so the director or feeder can turn and clear the aircraft onto their approaches, i.e. feeding the ILS with aircraft.

(Source: SentryRaven - Own Work)
The above picture shows the TMA airspace of Frankfurt (EDDF). The TMA arrival controller would clear aircraft onto the transition or a vectored downwind and hand them over to director.
The director position would then vector the aircraft onto the ILS for a handoff to tower.