This http://www.xavius.com/ordapp.html tutorial mentions the user work song as a final controller for two runways and says that during busy periods one controller would work each runway. I’ve never heard of this before and couldn’t find anting by googling on my own. So, what is a final controller and are they tower or TRACON?
"Final" or "Final vector" is the position in charge of, well, final approach. It is a radar control position worked in the TRACON.
As aircraft are transferred from the ARTCC to the Tracon, the Center (or Enroute) controller will tell them to contact the "Approach" controller. Everyone in the TRACON uses the callsign "Approach" (or "Departure"),* but the first person the pilot talks to in the TRACON will be the arrival position. This position is responsible for initial descent and sequencing of several arrival streams coming in to the TRACON's airspace.
After the arriving aircraft have been descended and slowed somewhat, they are transferred to the final position. This position is responsible for actually turning aircraft onto final approach course in sequence and with proper separation, and for monitoring separation along final approach to make sure the planes don't get too close. A full "PTAC"—position, turn, altitude, clearance—is quite a mouthful to say and eats up a good bit of frequency time, so during times of heavy traffic it is beneficial for there to be a dedicated controller for every one or two final approach courses; if they were also responsible for initial descent and sequencing they wouldn't have time to get out everything they needed to say. Having a dedicated controller at a dedicated radar scope also allows the scope to be zoomed in more, which increases detail and can allow for a tighter sequence because the controller is more confident that what they think is proper 3NM separation actually is.
*There is a very different type of radar controller who actually talks planes down to the runway using radar vectors only, instead of vectoring them onto an instrument approach and having the pilot follow a radio signal being radiated from the ground for guidance—this is an ASR or PAR approach. This position actually uses the callsign "XXX Final Controller" instead of "XXX Approach." Not all TRACONs have ASR/PAR approaches available; it requires dedicated training, and the controllers have to be sure to do so many per quarter to maintain currency with them, and for PAR approaches it requires specialized radar equipment. They are more common at military bases.
In addition to the above answers, the term final controller can also refer to the controller issuing instruction for a ground controlled approach, such as a PAR or ASR approach. Aircraft will get handed off to this controller before commencing the approach, and will receive lateral and/or vertical guidance depending on the equipment available for that runway. GCAs are almost exclusively limited to military airfields but used to be more common in civilian use.
A Final (radar) controller in a TRACON provides instructions (e.g., radar vectors, speed adjustments, etc.) and an instrument approach clearance (if required) necessary to sequence, provide separation with other aircraft, and align the aircraft with the landing runway. Once this is done the pilot will be instructed to contact the Control Tower.
Typically the aircraft will have been "handed off" to the Final controller by another TRACON controller who provides radar services a further distance from the airport.
At some TRACON locations, depending on the airport's runway configurations (parallel runways, two sets of parallel ruways, distance between parallel runways, etc.) there may be two Final controllers working separate runways.