A title to denote an R
-rated Preview for a movie.
When a studio releases a movie preview, or trailer, they get the MPAA to review it. Obviously you don't want A Nightmare on Elm Street preview in front of a Disney movie, so there are two types of previews.
The Green Band means it's approved for all audiences, and has G-rated material in it. You'll know it's Green because the screen right before the preview is Green, with the words
The following PREVIEW has been approved for
by the Motion Picture Association of America
Beneath that you see a rating for the film being advertised. Green means that theaters can show this preview in front of any movie, it's "suitable for all ages," ie. not scary or offensive.
Red Band trailers are different. They may have explicit content, as they can curse. They can have more sex, as the preview itself is basically rated R. The only rule is that a theater can only show a Red Band preview in front of an R or NC-17-Rated movie.
AFAIK, it costs something like $1000US to have the MPAA screen a preview and rate it.
Some special edition DVDs will show you both versions of the trailer, one clean the other explicit