This is the producer of a television show who has creative control over the content of the show.
The showrunner is almost always a writer, and manages the writer’s room, making the final creative decisions. How the plot unfolds, season over season, is the responsibility of the showrunner.
From an interview with the showrunner of Insecure:
You’re running the show – literally. You’re putting it together, from no idea to “Hey, this is what you’re now going to watch on the screen.” It’s hiring writers; it’s sitting with [the star] before we even have writers and arcing out where the season would go, how we’re going to explore these characters. It’s looking at casting and deciding who’s going to get the parts. And it’s being on set and [communicating] with the director and saying, “I need the joke this way.” Then, in editing, it’s revising all that stuff and doing music. It’s continuing to always be there.
This is required in television, rather than film, because the creative direction of a TV show unfolds over time. The show has to be steered over many seasons, and plots can develop throughout the show’s run that weren’t known when the show began.
Conversely, a film’s script is usually set before filming starts, and – except in rare cases where a sequel is already planned – the story ends when the movie ends. Once filming begins, there are few creative decisions to be made, in terms of the story.
As of this writing, the most famous showrunner working today is likely Shonda Rhimes, who runs (or ran) Grey’s Anatomy, Scandal, How to Get Away with Murder, and Empire.
If you like or hate a TV show, it’s usually due to the showrunner.