iTunes Extras are like DVD to Blu-ray movies extras that play right inside iTunes on the Mac or Windows, on the Apple TV, and soon with iOS 8, on the iPhone and iPad as well. They can include director's commentary, deleted scenes, bonus scenes, making-of documentaries, and other features.
iTunes Extras debuted in September 2009 and were available on iTunes desktop and the original, OS X-based Apple TV. When the Apple TV switched to iOS in 2010, it lost iTunes Extra support.
In July of 2014, Apple updated iTunes Extras. The old-style iTunes Extras were relabeled as iTunes Extras (Original) and stayed locked to iTunes Desktop. The new iTunes Extras immediately became available on iTunes desktop and on Apple TV with the 6.2 update. They were also announced as coming to the iPhone and iPad with iOS 8.
Not all movies that had the old iTunes Extras have the new iTunes Extras (hence the need to label them on iTunes desktop), and not all movies have iTunes Extras.