Star Trek wins again, fittingly (it's sort of surprising that they don't get so much as a nomination from here on, even considering the declining budget and quality of the show). "The City on the Edge of Forever" is an episode from late in the first season that's widely considered the original series' finest hour. A drugged Bones wanders through "The Guardian of Time" into 1930s America, where he saves a young woman's life. Kirk pursues him and falls in love with the woman, who is named Edith Keiller (Jackie Collins). Bones and Kirk soon learn that Keiller is a very persuasive spokesperson for the pacifist movement - now that she's alive, she will prevent the United States from entering World War II in time to defeat Nazi Germany. In other words, Bones is going to change history and destroy humanity's utopian future in the process.
It's a fascinating ethical dilemma, a wonderful character moment for Kirk (somewhat undermined by the fact that he falls in love with a woman in every episode), and a fantastic production all around, with most of the series' best acting on display. I'm not sure if it's my favorite episode, but it's up there, and I don't disagree with its status as a classic for a second.
I should also take this moment to mention that the screenplay was written by Harlan Ellison, a staple of '60s science fiction. Ellison won three Hugos for short stories in just four years in the late '60s. 1966 winner "'Repent Harlequin,' said the Ticktockman" is a psychedelic metaphor that protests the tyranny of schedules and time in modern society. 1968 winner "I Have No Mouth, and I Must Scream" tells the psychedelic story of five people tortured for eternity by a sentient computer he hates humanity for making him feel. "The Beast that Shouted Love at the Heart of the World" is just plain psychedelic. Can you tell it's the late '60s? I understand why Ellison is a legend, but his bleak and surrealistic visions really aren't for me. I think I'd rather read Ginsberg or Burroughs.
Ellison has long complained that Gene Roddenberry made several cuts and changes to the script (Star Trek mainstay Dorothy Fontana ended up with a co-writing credit). From everything I've heard about his script, the aired version sounds much superior (though the drug-dealing in the Federation sub-plot does sound intriguing). Ellison has a tendency to stir up trouble.