Seamless branching is a mechanism used on DVDs to allow the DVD player to jump to a different scene after finishing one. The most common purpose is to have several versions of a scene within one film, without having to store the entire film on the DVD several times.
A popular example is the DVD of The Lion King, where the user can select between the original cinematic version and an extended version. The two versions differ only in one scene (the "morning report"). The DVD player is instructed to play the film normally up to this scene, then jump to the appropriate scene as selected by the user before the commencement of the film, and then jump back to play the rest of the film. The user normally does not notice this jump, hence the word seamless.
Another possible use of seamless branching is for the localisation (translation) of on-screen visible text. Normally, only the audio track of films is translated into other languages, but when text central to the plot is visible on-screen, the scene may be shot once for every language, and the DVD player can be instructed to select the appropriate version of the scene depending on the user's language preference. (Unfortunately, even though the localised cinematic versions of Pixar's animated films such as Toy Story had much of the text in them translated, the United States DVD version contained the text only in English and did not use seamless branching for this purpose. In international versions, however, this was not the case.)
Always for this localisation purpose, the famous beginning text of The Hunt for Red October in the French Collector's Edition, both English and French screen-text are available depending on the asked language.