In the movie industry, the term "red-band" refers to the MPAA splash screen that appears before a trailer begins. While most that you see are green and "APPROVED FOR ALL AUDIENCES," some will be red and feature all the naughty R-rated content you'll find in the movie proper.
Naughty Dog's upcoming The Last of Us also has its own "red-band" trailer, which features more of the cussin' and limb-poppin' decaptitatin' violence that the sure-to-be-M-rated game will include.
This is the short 30 second version made for TV, as seen during last night's The Walking Dead finale:
At the end, it points you to Amazon, which is hosting the official 2-minute red-band trailer. Of course, the internet being what it is, someone has already uploaded it to YouTube:
And unfortunately, the video has been removed. Thanks, ESRB!