I don't know if these posts will be annoying. The thing about posting less frequently is, there's something like five hundred episodes of Star Trek, and I'm not getting any younger. I also hope that—activity begets activity.
This is continuing to follow production order rather than air date, hence the third episode is "The Corbomite Maneuver." It was the tenth episode aired.
Synopsis
The Enterprise is waylaid by a cube, which threatens to kill them with radiation. They destroy the cube, and decide to proceed forward—after all, their mission is to seek out new life, even life that sends out obnoxious cubes.
They're soon set upon by a gigantic sphere that threatens to destroy them, but gives them ten minutes to make peace with their impending deaths. Spock compares their situation to chess, where they've been mated... in passing, maybe. I dunno, Google it.
Kirk decides to play poker instead, and says that all Earth ships (there's no UFP at this point) are fitted with retaliatory superweapons called corbonite devices. This fictional device reflects all attacks, destroying the attacker. The sphere asks for proof, and Kirk says no.
A smaller ship breaks off to tow them to a habitable planet, where it intends to maroon them. Kirk plays chicken with the smaller ship's tractor beam, noting that it heats up their engines but drains power from the other ship. The Enterprise wins, and the small ship powers down.
He beams over with McCoy and Lieutenant Extra—by the way, this is the first episode with McCoy—and find the alien captaining the ship, all by himself, is lonely. The alien wanted to make sure of their intentions before letting them aboard. Kirk leaves Lieutenant Extra behind as a cultural liaison.
Commentary
This episode established another enduring Trek archetype: the monster of the week that turns out to not be a monster. Seeming dangerous isn't the same as being dangerous, and sometimes the apparent danger just wants some understanding.
Also, Kirk's most important character trait is established here, revealing why he's captain rather than Spock. Kirk refuses to lose, and he'll do anything to keep from losing. This is most famously explored in The Wrath of Khan and the Kobayashi Maru scenario. It was designed to be unwinnable, to test the reactions of potential Starfleet officers to an impossible situation. Kirk is one of the only people to have beaten (?) the simulation, and Spock's protege Saavik asks him how he did it. It's simple, he says, I cheated.
It's worth taking a moment to appreciate what a leap forward this was in televised science fiction.
It feels like they were 90% of the way through shooting, then realized that the big rubber puppet they had for the viewscreen alien wasn't gonna work out for, you know, more than a static shot with vague mouth movements. Then in a panic they brought on Howard and got him a chaise longue and called it a wrap.
I doubt it actually went down like that, but it's a nice thought.