Plain English Meaning
The controller who takes over from the tower right after you take off and guides you on radar until you're high enough and far enough out to be handed to the next controller.
Full Definition
The radar-equipped ATC facility responsible for separating and sequencing IFR aircraft after takeoff, from the moment the tower hands them off until they are handed off to the en route center (ARTCC). Departure Control is a function of the Terminal Radar Approach Control (TRACON) serving the airport.