From the original series: The Doomsday Machine, for its razor-sharp dramatic tension and timing:
[Full episode, no embedding available]
From The Next Generation: The Inner Light, in which Jean-Luc Picard experiences an entire lifetime on an alien world in the blink of an eye.
From Deep Space Nine: Duet, when Kira Nerys encounters a Cardassian trying to atone for the sins of his people.
From Voyager: Endgame, the conclusion to the series. I like Captain Janeway, but I love Admiral Janeway for her sheer ruthlessness.
I've only seen one episode of the animated series and about a dozen from Enterprise, so I have no favorites there.
What are your favorite episodes of Star Trek?