What is the differences between Dominant, Marketing and Operating Airlines?

 
The marketing airline is the airline that sold the ticket, and the operating carrier is the carrier that actually carried the passenger.  You'll mainly see this distinction when you're looking at regional / express carriers, like SkyWest (OO) flying for UA or similar. 
 
We use the term "dominant" carrier when we’re dealing with itineraries where more than one carrier is involved, and we have to assign the itinerary to a single carrier.  
 
In DB1B data "dominant" carrier is determined based on the maximum single segment mileage. In all other products "dominant" airlines are determined based on total segment miles by a carrier.  Only on rare occasions will it change the dominant designation.