Short answers, pulled from the story.
Continuity in fiction refers to the consistency of characteristics of people, plot, objects, and places seen by the audience over time. This concept matters most in long-running stories where audiences invest years into following a single narrative thread.
Script supervisors gather paperwork, photographs, and documentation to note details for maintaining the continuity of the production. These records capture factors within the scene and technical details like camera positioning and equipment settings to ensure related shots match even though filming has been split up over months on different sets and locations.
Menelaos kills a minor character named Pylaimenes in Book V yet Pylaimenes remains alive to witness the death of his own son later in Book XIII. Roman poet Horace coined the phrase regarding this phenomenon in his Ars Poetica lines 358-359 stating that even the great Homer nods off occasionally.
Producers use rapid aging to open up characters to wider ranges of storylines and attract younger viewers despite the resulting timeline conflicts. The BBC's Merlin series features Mordred initially played by a young child in Season 4 who suddenly grows into his late teens by Season 5 while other characters aged by only three years during that same period.
Creators sometimes deliberately employ continuity errors to generate humor or achieve specific artistic goals rather than through negligence. In the Marx Brothers' classic film Duck Soup, Groucho Marx speaks a line followed by a cutaway shot then another shot where his hat changes to something more outrageous like a Napoleonic hat.
A conflict with previously established facts can be deliberate and is called a retcon which stands for retroactive change in continuity. When holders of intellectual property discard all existing continuity and start from scratch, it is known as rebooting.