A side story is a story that occurs alongside established stories set within a fictional universe. As opposed to a prequel, sequel, or interquel, a side story takes place within the same time frame as an existing work.
Side stories are common in epic type series, especially war oriented series where it is possible to tell many stories from many different points of view. Early examples of a side story are found in the ancient Indian epics Mahabharata and Ramayana, which contained numerous side stories which were loosely related to the main story of those epics. It is typical for side stories to be self contained, small scale events, insignificant in the bigger picture. They tend to be one shot stories with a beginning, middle, and end and focus heavily on character drama while the major action occurs mostly in the background. They are commonly used in sitcoms to keep interest in the story or to use remaining time.
A side story is not quite the same as a spin-off. A spin-off takes already known characters, usually supporting or background characters and involves them in a story or series which highlights them and further develops their character. The series' true main characters may make cameo appearances or be referred to in dialogue. Side stories, rather, focus on a completely new set of characters who have no history and typically no connections to the existing main characters. The settings for side stories are intentionally away from the major events that the main characters are known to be participating in.
The term "side story" is widely used in anime fandom, where it is a direct translation of the Japanese word gaiden which is often used to describe such stories.