Redubbing

The practice of removing all the dialogue from an existing movie or television show, and replacing it with an entirely new recording. Most common, of course, when translating a show for audiences which don't speak the original language.

This poses few problems when dubbing an animated feature, such as an anime or Disney film. Unfortunately, live action stuff sometimes results in dialogue which fails to flow naturally or is subtly out of sync with the character's lip movements, causing many fans of imported films to (violently) prefer the original language version, relying instead on subtitles to understand what's going on. At worst, it can be hilarious -- take, for the perfect example, the Godzilla films.

Sometimes, rather than attempt to recreate the original dialogue, the copyright holders will simply replace the entire thing with new dialogue, to humorous effect -- a Gag Dub.

Anime

 * Any anime with a "dub track" - thus, too many to list here.

Film - Animated

 * In the 2006 dvd re-release for An American Tail in the scene where the three orphans torment Fievel, two of their voices were redubbed the fat one and the short one, originally they were voiced by children in the redubbed dialogue the fat one is given a low gravely voice and the short one is given a nasally voice both by grown men.
 * The 2005 British CGI film, Magic Roundabout who redubbed in the US as "Doogal" and replaced all but two of the British celebrities who provided voices for the characters with American ones.
 * And of course the original series of The Magic Roundabout, which took a French show and redubbed it with a completely new script.

Film - Live-Action

 * Any number of martial arts movies and parodies of the genre.
 * One of the most pointless redubbbings was in Arthur, in which Arthur's new voice actor redubbed the season during which his original voice actor had gone through puberty. Especially jarring for viewers who had viewed the episodes before the edits.
 * Weirdly, The Gods Must Be Crazy was redubbed for North American release even though the actors who weren't Bushmen spoke English. It was thought that their South African accents would be difficult to understand.
 * Godzilla: Though played (mostly) straight, these films tend to be hilarious to English speakers, as the translated dialogue is often multiple seconds out of synch with the original lip motion.
 * Andie McDowell's lines in Greystoke: The Legend of Tarzan, Lord of the Apes were redubbed, by Glenn Close.
 * Arnold Schwarzenegger's first feature film, Hercules in New York had all his lines redubbed by an actor without the Austrian accent.
 * One of the most notorious instances of this happens in the unintentionally hilarious 1968 drama The Legend of Lylah Clare. It seems actress Kim Novak could not handle the German accent, so another actress with a much deeper voice dubbed on the Lylah voice when required. While this did indeed make it seem like her meek actress character was possessed by the ghost of the dead Lylah, it's also very obvious dubbing. Made all the funnier by the heavily Italian-accented Rossella Falk as Lylah's former dialogue coach!
 * The Life and Death of Peter Sellers had a sad example. After he and his wife separate, Peter redubs his wife's lines so that it sounds like they stayed together.
 * The initial American release of Mad Max had all the dialog redubbed by American actors, without the Australian accents.

Western Animation

 * Parodied in an episode of The Simpsons. The retirement home has a redubbed Gone with the Wind where Scarlett & Rhett stay together and all the American Civil War parts were removed.
 * The Thomas the Tank Engine fan community is notorious for doing this, but occasionally end up doing an (arguably) better job than the original narrator, mostly due to the fact that many of these fan redubs have /very/ large casts.