As per the exact question posed in this thread, these are NOT my favoriate episodes per se…but rather each of those are what I say is representative of each series.
Star Trek: The Original Series
Taste of Armaggedon (I just watched this yesterday actuall).
This has everything:
Kirk and Spock breakout out of a lockup.
Spock neck pinch with humor.
Kirk kick ass with hand to hand action.
Kirk decide a planet’s culture isn’t growing so therefore must change.
A moral message: War is not clean. It’s horrible and should feel horrible. Wars can go on forever unless leaders decide enough is enough.
Scotty being awesome. McCoy being awesome.
A douche-bag Fed ambassador that is self-righteous but gets proven wrong.
Ship in danger. A new planet to contact.
Red Shirts in action. (But, sorry, no redshirt deaths this time).
No Kirk shirt ripping (uh oh).
Star Trek Enterprise:
Strange New World
This episode really got to show what it was like before the days of Kirk, when Star Fleet was naïve about new worlds.
They don’t take any real precautions. And Archer even brings his dog down and lets it pee on the planet. (Spock would be horrified.)
Star Trek: The Animated Series
The Terratin Incident
This is representative of the animated series because it could do things physically that a live action show —particularly in 1973 couldn’t do.
In this one, the crew of the Enterprise keeps shrinking smaller and smaller. And the encounter a planet with a tiny civilization of tiny people.
Star Trek: The Next Generation
Who Watches the Watchers? (This also either my favorite TNG or tied for fav with Measure of a Man).
This episode is representative because it shows an encounter with an alien species that you end up really caring about. And you get why the Prime Directive matters. And you also see that Picard very literally, would risk…even give …his life in the service of dispelling the myth of a high power “god”. TNG is a thoughtful show. And this is a very thoughtful episode.
Deep Space Nine
The Search (Part 1 and 2).
This is when the DS9 crew are pulled into the larger world of the Dominion. And we get to meet the founders. The struggle with the Dominion is so central to DS9, that this is the representative episode for me
Star Trek Voyager:
Equinox (Part 1 and 2)
We get to meet the Federation Starship Equinox who is also lost in the Delta Quadrant.
By seeing another ship that is so much worse off, we get to a better appreciation Voyager, it’s crew, and the choices Janeway has made.