This is how it was explained to me when I was testing for FDNY dispatch, I never became one so you know.
Alarm Receipt Dispatcher - These are the call takers and ask certain questions to ascertain what the incident is and where. They enter it into FDNY's proprietary CAD system.
Decision Dispatcher - Looks at the information from the CAD and the units it recommends to send to the incident. The DD along with the Tour Supervising Dispatcher can modify the units responding ie sending another unit if one is not available. Once they "release the call" the information is sent to the firehouses and MDTs.
Voice Alarm Dispatcher - They are the ones who announce the incident over the firehouse intercoms and also verify any acknowledgment of alarms by the units at the firehouses.
There were two Radio Dispatchers, In and Out. The Radio In answer units calling the borough Communications Office. The Radio Out announced all the incidents over the radio and also would call units in the field to relay information, etc.
In charge is the Tour Supervising Dispatcher aka the Shift boss. Depending on the borough he / she oversees 4 to 8 dispatchers. Also some of the positions could be combined depending on the borough and time of day. The two radio dispatcher positions were one position that could be combined, at least when it was explained to be.
There are a few members here (active and retired) that can give you a more in depth explanation how it works now.