The Continuity Cap was a time device created during the Dark Times by a race called the Hyffons, who hoped that it would bring a war on their home world to an end. When worn by the user, the cap would show them moments from their personal past and allow them the option of changing previous choices or events. The cap then would then hunt for a parellel universe where the chosen alternations naturally took place, and would swap the elements between the two worlds. This often had bizarre side-effects on reality.
The Hyffons were able to change their planet's political history to stop the war from ever existing, but it fell into the wrong hands. Ultimately, the race "disappeared into a puff of its own continuity."
Stephen Cropper found the Cap in a Bed and breakfast in Cardiff (which the Doctor referred to as "a Mecca for absent-minded aliens"). Coming to understand its purpose after putting it on, he chose not to use the cap as he believed that altering his own past would lead to him never finding it, causing a paradox. He instead started a business and claimed that cap to be his own invention, offering to allow others to change their past for a specific price. This resulted in the 21st century becoming an eclectic mix of alternate universes and events, with every other universe also being effected by "Earth's history being shuffled like a deck of cards."
The Tenth Doctor investigated the situation after finding his own timeline to be effected by the technology. He confronted Cropper, deduced the true nature of his "invention", and convinced him that these events needed to be unwritten. The Doctor instructed Cropper to put on the Continuity Cap and change his reaction to finding it in Cardiff; instead of trying it on, he instead would put it in a package and mail it to the Doctor. As this took place, reality was put back to normal, and the Doctor and Donna left to retrieve the Cap in Cardiff. (COMIC: The Continuity Cap)
Behind the scenes
The Continuity bomb is fundamentally similar to the Continuity bomb from the story Four Doctors.