One of the most justifiably, universally praised episodes in the entire franchise, "Duet" is a fascinating exploration of the limits of moral responsibility.
Kira becomes involved in the case of a Cardassian accused of being one of the worst war criminals of the Occupation. Being a lifetime member of the Resistance, she can easily be identified as someone with a vested interest in seeing Cardassians get what's coming to them. She was a terrorist, freedom fighter, and patriot all rolled into one.
Except life on the station and in the aftermath of Bajorans regaining control of their own lives has complicated her worldview. The old Kira probably would have assumed the Cardassian was guilty and never given it a second thought, condemned him on the spot. Yet a man who is doing everything to accept his fate forces Kira to reexamine everything she's assumed, not just in his case but throughout her entire life. Something just isn't right.
I don't want to spoil "Duet" for you if you've never seen it, but suffice to say it's one of the most complex episodes of any Star Trek. There have been numerous other, very similar efforts, from Next Generation's "The Defector" to Voyager's "Jetrel," and perhaps even Enterprise's "Stratagem," but "Duet" still holds the high-water mark for piercing ambiguity. Kira's journey during it is not only a defining one for her, but for the entire series. In short, this is the one episode that finally cracks the formula of what Deep Space Nine was supposed to be like from the start.
Maybe you want to think of "Duet" in that big sense, and maybe that's a little much, and may actually explain why so few Star Trek fans have rallied around the series. Despite the fact that those who love it really love it, Deep Space Nine still has the same amount of overall approval as the two later series that most fans will only admit to hating. With the original series, and certainly for its time (even as sci-fi television was perhaps best defined by The Twilight Zone, a series Deep Space Nine might be said to be patterned after), good was good and evil was evil, and it was unusual to blur the lines of these distinctions. Next Generation kept that model for the most part (hence the signature, faceless foe known as the Borg), but right from the start, Deep Space Nine took a different tack, and "Duet" is the moment where everything came together.
For a young series, it's an incredibly mature episode. It's arguably the moment Kira stopped being a surrogate for the character she was created to replace (Ro) and instead came into her own. Yes, it's another moment in the first season where she's forced to question her assumptions, but it's the first time she's doing it with a Cardassian, looking at an issue and suddenly finding a whole new perspective. That's what most of the characters in the series struggled to find, and "Duet" is the first time one of them is successful.
Whatever reason you yourself will find, "Duet" will leave an impression on you. It's Deep Space Nine in a nutshell.
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Memory Alpha summary.