A spin-off is a devlopment, often in another medium - like Lara Croft starting out as a video game and then having a movie 'spin-off'. A sometimes related word is 'tie-in', where the spin-off is already planned from the beginning (like a book with a TV tie-in).
I know as little as possible about the video game world, but I assume there's a company called Gearbox with a product called Duke Nukem Begins, spun off from a previous game (possibly called 'Duke Nukem' ... ). In that case it would be what is also known as a 'prequel' (like the movie Batman Begins).
Spin-offs, tie-ins, prequels... It's all about wringing the last possible drop of profit out of a basic idea