I'm glad someone mentioned the animated series. Yes, the animation is pretty lousy even for 1973-74, but you had the original actors, a lot of the original writers . . . and "The Slaver Weapon." Based on a short story by SF writer Larry Niven, who adapted his tale for Trek, it features the catlike aliens, the kzinti; a strange weapon found in a stasis box in which no time passes; and no Kirk or Enterprise!
The rest of you have mentioned a lot of my favorites. I'll just toss in ST:ToS's "Corbomite Maneuver," with some of the most vivid dialog the show ever produced, and "Charlie X."
As for TNG, there were so many great stories. "Inner Light," "Darmok," "Measure of a Man," "Q Who?" (the introduction of the truly alien and terrifying Borg), "Starship Mine" (a kind of "Die Hard" aboard the Enterprise), and the story featuring Beverly Crusher in which she murmurs two of the neatest lines ever:
"If there's nothing wrong with me . . . is it possible there's something wrong with the universe?"
and
"Click my heels together three times and I'm back in Kansas. Can it really be that simple?"