I think I'd suggest a planet-of-the-week episode. From TOS, "This Side of Paradise" or "A Taste of Armageddon." From TNG, maybe "Who Watches the Watchers" or "Up the Long Ladder."
It's such a great example of what Picard is about, what makes him a great man, as well as being a great commentary on that inner human struggle we experience on both a personal and social level.
I came here with exactly this episode in mind. I think it is representative in a few ways:
It involves an alien of the week.
The alien species is culturally similar to human societies we, as viewers, are familiar with.
It demonstrates what the Federation is all about, including the Prime Directive, respectfully dealing with less developed civilizations, and solving problems without violence (especially when the problems are your own fault).
It's more or less self-contained. Whether this is "representative" is debatable, I guess. I think it's a big part of Star Trek even though there's a larger focus on season-long storylines in later series.
"Who Watches the Watchers" is self-contained but you can also show someone "The Drumhead" after they've seen it. Beyond that also being a good episode for newbies as suggested elsewhere in these threads, the callbacks to "Who Watches the Watchers" nicely illustrate how threads from one story can keep affecting and enriching the show's universe.