I haven't seen many of these for years and years though one of my stations is currently airing the crappy redone versions so I get to refresh my memory. I don't remember all of them and don't seem to remember the third season much at all. So I definitely reserve the right to change my mind.
Everybody's already mentioned most of my favorites from season 1: "Arena" (Kirk vs. Gorn) got a mention, "The Devil in the Dark" (one that may be better in the thinking about than in the watching, but which is really great -
not a bumpy forehead), and "This Side of Paradise" (drunk Spock playing in a tree) all got a mention. (I think that was a 'best' and not a 'worst' on "This Side".)
My very favorites are the consensus winner "The City on the Edge of Forever" (6 mentions), "Balance of Terror" (3), and "The Menagerie" (2). I love "The Menagerie", not so much for itself, but for what it shows about how Star Trek could have been - closer to the original vision of what could arguably have been an even better show. A female second officer in the same almost DS9-like uniform everybody else wore, for one example. There is something simultaneously more advanced and more classic about it. ST was a very 60s show. The ST that might have been had strong 50s aspects and strong "now" aspects. And it's hard to argue against "City" - but I really think I might give the edge to
"Balance of Terror". As was mentioned about "Arena", it's lacking in some production values in some spots - a couple of things are pretty silly (the viewscreen works an awful lot like a camera and a director rather than... y'know, a viewscreen) but it's a fantastically tense, balanced, exciting, yet philosophical and well characterized episode.
The one particularly noteworthy s1 ep no one mentioned (unless I missed it) is "The Enemy Within", one of the many "split" episodes and probably the best. Kirk's animal and rational sides split. I really like that one, though it's not up to the very best.
I don't think there's anything in s2 to challenge s1 but I like "Amok Time" (Spock in heat - 2 mentions) and "Mirror, Mirror" (3 mentions). It's a peculiarity with the Star Treks that I love the three series but some of my favorite episodes are the "alternate worlds" eps - those that run contrary to the series. My favorite ST of any series, all-time, hands down, is "Yesterday's Enterprise" (TNG - a temporal rift causes the Enterprise C to not be destroyed saving Klingons, so the Federation and Klingons are at war) and I really enjoyed the Intendant-universe DS9s.
No one mentioned "Catspaw". Freaky Goth ST with a compelling performance by Antoinette Bower and a memorable ending. Or "Wolf in the Fold" which has Scotty, of all people, accused of murder and puts Earth history to interesting use. Or "By Any Other Name" which has the Andromeda folks try to take over our galaxy and puts Kirk's romancing of the alien to better than usual use. But these are only particularly memorable and not necessarily candidates for "the best".
And, as with many others, hands down, no contest - the worst is "Spock's Brain" although I agree with Daisy-Boo's take at the same time.