These devices are predominantly seen on episodic series where every installment furthers the plot, but otherwise are used for plot arcs, that is, when an episode is going to pick up on a storyline intitiated several episodes ago. A "Previously" will almost always be present in an episode that picks up from a "To be continued...".
These montages are inserted as the first thing into an episode so that viewers who had not seen the previous episodes or who do not remember what happened can understand from where the current episode will begin. They usually begin with a voiceover proclaiming either, "Previously on... [the show's name]", "Last (time/week/season) on... [the show's name]", "Earlier on... [the show's name], or "On the last (episode/season of)... [the show's name]". Many shows have begun to use a main character's voice for this voiceover rather than a neutral narrator, and some series such as Stargate SG-1 and Stargate Atlantis alternate which main character says "Previously on Stargate Atlantis...".
On the Fox show 24, the main character, Jack Bauer's voiceover "Previously on 24..." is followed by scenes from prior episodes of the current season, not necessarily from the very last episode, that concern major plot points of the current episode. This was carried over into 24: The Game, where, once a player completes a mission, all cut scenes prior to that mission can be viewed in the manner of the tv show. The playable scene are not shown, as the player is given the opportunity to replay them to get a higher score.
In the English dubs of various episodic anime series that suffer editing for time or censorship reasons, recaps are sometimes extended or introduced where they didn't exist originally, to cover any shortfalls in running time. Some examples where this occurs are Robotech, the FUNimation English dub of Dragon Ball Z, and the 4Kids English-language version of One Piece. The original Japanese version of One Piece did not always have recaps, though many episodes did have recaps as the first scene following the opening credits.
There is some debate as to whether these recaps should be considered as part of the episode whole. Some home video and DVD releases of shows that use recaps include them, some do not, and some seem to randomly use or omit them for different episodes within the same collection.