While Star Trek is often dismissed as a weird subsection of nerd culture, it's hard to downplay how hugely important it was to society (outside of the convention world) - the series, the brainchild of creator Gene Roddenberry, presented a vision of the future that was almost unheard of at the time (near the height of the Cold War): it was a future that was bright, where the possibilities were endless. It was a future where people of all beliefs, nationalities, creeds, races, and species worked together to achieve great things. It was a future of tolerance and peace. While the rest of the world was expecting the future to be one of burnt skies and atomic fallout, Star Trek said we'd be traveling through space (granted - to 1920's gangster planets and also wrestling with lizard people, but still).
Also, it contained the first scripted interracial on-screen kiss in television history, between Nichelle Nichols' Lieutenant Uhura and William Shatner's Captain Kirk (granted - while both were being controlled by faux-Greek aliens using telekinesis, but still).