Continuity editing involves shooting a scene with multiple shots that integrate seamlessly together to maintain an uninterrupted flow and focus for the audience. It can involve scenes shot at different times edited together or varying short scenes to keep the movie progressing without dwelling too long on any one scene. Examples of continuity mistakes discussed include discrepancies in background props, clothing, lighting, and vehicle positioning between shots within the same scene. Maintaining continuity is important for verisimilitude and audience engagement with the story.