In television industry parlance, "showrunners" are the people, working either independently or as part of a small team, responsible for a television series as a whole.
Compared to movies, in which a single director has significant control over the filming of a single story, usually working with a small group of primary characters played by experienced actors, television is a bit more complicated. Each episode of a given television series might be scripted by a different writer, shot by a different director, and is more likely to be acted out by a broader cast with less experience under their belts. Despite all of this, viewers expect shows to exhibit a more or less consistent tone and artistic posture, and in the case of dramatic shows, to develop plots and themes which last several episodes, if not seasons. It's the job of a showrunner to make that happen.
Showrunners, experienced television writers with a feel for good scripting, might be brought on board by a production company looking for a steady hand at the tiller of their latest project, or they might get the position when one of their ideas is picked up for production. On the job, showrunners are their series' gods. They head up a show's writing team, coordinate story threads, polish staffers' scripts, write a few of their own, and massage the collective result into a coherent style. They're often responsible for creating and maintaining the series "bible", the master collection of series background and guidelines for writers to work from. Showrunners have significant control, either directly or through delegates, over casting, location selection, and even directorial tasks such as visual style and character interpretation. Though usually writers first, some showrunners may direct episodes of their shows, and in cases like sitcoms focused around a standup comedian, might be actors themselves. For all this work, showrunners are usually credited as "executive producer". Not all executive producers, however, are actually showrunners; some are closer in job description to film producers, concentrating on the funding, legal, and business end of the operation.
When a showrunner does his job, you get decent television. When he does it well, you get series that win critical praise and high ratings, bringing in the dough for their production companies and broadcasting channels. Though you might not remember the name of your favorite series' showrunners off the top of your head, you'd probably recognize some notable ones – Aaron Spelling, David E. Kelley, or Joss Whedon, for example. These showrunners occupy roughly the same place in television cosmology as famous directors like James Cameron, Stephen Spielberg, or Quentin Tarantino do in its movie equivalent, able to make or break a deal by their name and reputation alone, trusted by studios and viewers alike to deliver a product they can depend on.