Kirk. Times two.
This is the episode where the concept of transporter malfunctions becomes a Star Trek trope. It also sets up the idea of decontamination later featured in the prequel series Enterprise. In essence, it also sets up the idea of the Mirror Universe later featured in "Mirror, Mirror" and numerous other franchise episodes.
"The Enemy Within" is also the first character study episode in the franchise. I don't want to rate its importance too high, however, so I list it mostly as a character episode for Kirk, in which his good and bad sides run amok, the bad side featuring his worst romantic impulses (certainly interesting for this character, and worse if you happen to be Yeoman Rand).
If your idea of a good time is William Shatner ranting for an hour, then this is your kind of episode. (Which might also explain why the later Mirror Kirk is barely seen.) If you want the episode that might explain why people think Star Trek features mostly bad acting, this might be the one they're thinking of. It doesn't reflect exceptionally well on Shatner. But I guess it doesn't particularly matter.
Please note that for the curious record, Uhura does not appear in this episode. In fact so far Rand seems to have been more important to the series. Of course, this trend rapidly reverses itself. By the end of the series, Rand will have become a blip, and it's Uhura who gets the last laugh with a far more historic smooch with Kirk!
franchise * series * essential * character
Grace Lee Whitney
Memory Alpha summary