Continuity is the maintenance of continuous action and self-consistent detail in the various scenes of a movie, or any other audio-visual media. The crew person in charge of keeping continuity is a Script Supervisor, (also called continuity supervisor), who oversees the continuity of the film, including wardrobe, props, set dressing, hair, makeup, and the actions of the actors during a scene. And, sometimes the continuity mistake lies in the mis-match of time periods, as you can see in the image below.
Films and even corporate video are mostly shot out of sequence, so continuity is very important to keep up the uninterrupted flow of visual storytelling. This is why, shooting with sunlight (moving patterns on a wall), or wearing a shawl (tends to have a life of its own), is not advisable.
Here some classic continuity mistake examples: