The one with the guy who battles himself. Wait, let me clarify: the one with the guy who is not Kirk who battles himself.
The "guy" is Lazarus, who comes to Kirk seeking help. He has awesome facial hair. Lazarus, I mean. Kirk was always clean-shaven (except for the stubble in The Undiscovered Country).
Most significantly about "The Alternative Factor" is the first time in franchise lore where alternate realities are featured. There's a lot of sci-fi hooey involved in the story. You don't have to worry too much about the details. Bottom line is, this is an episode that is nearly completely stolen by the guest character.
And that in itself is notable. Sometimes "Alternative Factor" is viewed as the series at its most extravagant, most preposterous. And it basically is, really. This is Star Trek going for broke, and in its own way is really a sight to see. Not one of the most famous episodes, but perhaps one of the earliest of what would become known as an equally infamous streak that would help give the franchise its reputation as something hard to take seriously. Viewers at the time, basically, probably had this one in mind, rather than the classics surrounding it, when they forced NBC to hedge on keeping the series around.
But yeah, it's also kind of awesome, an episode far easier to appreciate when you don't have to worry about the fate of the whole thing. Just Lazarus.
franchise * series * essential * character
(Read a nearly completely different version of this episode that almost happened here.)