Riding a wave of nostalgia recently for James T. and the original phasa gangstas . . . but I know there are at least a few episodes I’ve never actually seen (e.g., “Amok Time”). Enhance my Netflix queue, Trekkie Dopers: your desert island list of the orig. episodes?
(The only one I can identify by name as having seen–and more than a few times–is “City on the Edge of Forever.” Am still in awe of it every time I see it.)
My personal favorite is “Errand of Mercy,” the episode that introduces the Klingons to the Trek universe.
Kirk and Spock try to intervene when the planet of the pacifistic–and strangely complacent–Organians is taken over by the Klingons. I always enjoy how frustrated, and even contemptuous, Kirk gets while trying to stir up a rebellion among the Organians, who just don’t seem to care about being invaded. The ending, where he and the Klingon captain are practically buddy-buddy when they argue about their right to get into a war is priceless.
The Troubhle With Tribbles–fun, well written, & gives everybody a good line or two.
Uhuru gets more lines here than in any other episode, I believe.
Er, would prefer only the vaguest of summaries, **Miss Mapp. **I’m old-fashioned with when it comes to spoilers. 
“City on the Edge of Forever,” just for Kirk’s “Sophie’s Choice” dilemna.
Second place goes to “The Trouble With Tribbles,” for sheer laugh-out-loud comedy. Nothing Harry Mudd does can compare to dumping a million tribbles on Kirk’s head. 
“Mirror, Mirror” - The Counter-Universe episode, and Spock with a beard!
“The Trouble With Tribbles” - Just plain fun.
“The Enterprise Incident” - Romulans and treachery and good stuff like that.
“Journey To Babel” - Spock and his father do some bonding.
“A Private Little War” - If for no other reason than Spock’s concluding line!
Oops. Make that last one “Friday’s Child”
Ok My Trek must see List
Charlie X-- Early Star Trek with a nice creepy ending.
Balance of Terror-- First real villians for The Federation are the Romulans
Errand of Mercy – Klingon’s first introduced
I MUDD - best Harry Mudd episode
Trouble with Tribbles-- Classically funny
The Tholian Web – Great McCoy Spock Dynamic, Interesting Peril.
City on the Edge of Forever – Nuff said
Arena – Kirk battling a giant rubber Lizard works better than it should
Mirror Mirror – Loved the Mirror Enterprise.
Space Seed – KHAAAAAAN!
and for a hokey good time
The Way to Eden
Spock’s Brain
Both are goofy but memorable in their sheer badness.
Most of my favorites have already been mentioned, but I’ll add Devil in the Dark. Spock gets to mind meld with a giant, acid-oozing slug and McCoy’s classic “I’m a doctor, not a plasterer!”
“Mirror, Mirror”
“The Trouble with Tribbles”
“This side of Paradise”
“The Immunity Syndrome”
“Amok Time”
“City on the edge of Forever”
If you want to laugh at the original series instead of with it (which can be as entertaining and in no way makes it any less of a great show), watch Spock’s Brain, unanimously voted the Worst TOS Episode Ever by the general consortium of worldwide Star Trek fans.
I’ll add only The Menagerie, Pts 1 and 2 to the list of great episodes already posted.
Most of my faves have been mentioned - Tribbles, City, Devil in the Dark, Enterprise Incident, Piece of the Action …
But not Spectre of the Gun. I love Spectre of the Gun.
“Brain? Brain? What is brain?”
I can’t hear you! I can’t hear you! LA LA LA LA LAH!
Plus one extremely hot Romulan commander in a really tight dress.
Bricklayer. He wasn’t a bricklayer.
*For The World Is Hollow And I Have Touched The Sky
The Paradise Syndrome
The Enemy Within*
One McCoy vehicle, two Shatner… All three have them going beyond their character’s normal bounderies. Plus all the other ones listed.
Face it, you’ll end up having to see every last one of them, good or not, because you are now hooked.
Even the shows that are so terribly bad end up being fun to watch simply to work on your nitpicking skills. For these, try:
*Operation: Annihilate!
The Alternative Factor
The Apple*
The black/white refugee episode.
The submarine vs. destroyer episode (based on Burt Lancaster’s Run Silent, Run Deep ?) Enterprise vs the Romulans and their cloaking device. The Romulans even have a periscope, too!
Many years ago I had a friend who wanted to write/see an operetta based on ST:TOS. He’d only come up with one line, though, and it was for McCoy, to wit:
I’m a doctor, not a counter-tenor!
I have to add “The Doomsday Machine” and “The Immunity Syndrome” for their space-monster-battling, scenery-chewing goodness. Watching William Windom ham it up with Shatner is a riot.