It's a Haruhi Suzumiya spin-off, set in the alternate universe from the movie and fourth novel, The Disappearance of Haruhi Suzumiya. (It even begins around Christmas, picking up the baton from where the movie left off.) I really enjoyed it. It's gentle and mostly inconsequential, lacking the original's frenzied high-octane SF ideas and storylines, but for me it's the most likeable entry in the franchise.
The original Haruhi Suzumiya series of course involves an alien robot (Yuki Nagato), an esper (Itsuki Koizumi), a time-traveller (Mikuru Asahina), a long-suffering human (Kyon) and a megalomaniac bullying freak (Haruhi Suzumiya) who doesn't know she's a reality-warper who could accidentally destroy the world just by getting in a bad mood. Haruhi keeps making their lives hell, because she gets bored.
This premise was a novel series, which then got adapted into manga, video games, radio dramas and a famous anime. One of the more eccentric spin-offs was a 2007 four-panel parody strip that also got an animated adaptation, The Melancholy of Haruhi-chan Suzumiya. The artist's name was Pyon. He's since also created non-comedic spin-off manga, the main one being The Disappearance of Nagato Yuki-chan (2009+, ongoing) and it's now become an anime.
In its universe, everyone's a normal human. No one has reality-warping powers, although otherwise they're basically the same as usual. Yuki's the main exception. Here she's just a shy girl who's in love with Kyon and can't bring herself to say so. Everyone else, though, is immediately recognisable. Kyon's still the ultimate straight man, sardonic, sarcastic and not suffering lunatics gladly. Haruhi's still chasing aliens. Mikuru's still pathetic and Koizumi's still a smiling yes-man who follows Haruhi everywhere and agrees to everything she says.
It's the old gang, in other words. It's just that here they're in a calmer, gentler show in which little of importance happens, with the focus on their relationships.
What makes it particularly intriguing, I think, is its awareness of the original. The nods are mostly occasional and subtle, but this isn't a show that's blindly doing its own thing. It knows it's an alternate universe and that there are other versions of everyone. They're still the same people. We know how they feel. We know who's secretly in love with whom, even if some of what we think we know has been rewritten. The main one is that Kyon no longer fancies Mikuru, since this series is essentially a what-if fanfic for Kyon-Yuki shippers. It's possible to justify this, mind you. One might speculate that Kyon has strong protective instincts and there's no need to protect Mikuru any more since she's no longer being bullied by Haruhi. Instead the wounded duckling is Yuki, so it's Yuki he ends up hanging around.
Thus we know from the parent show that Haruhi demonstrably likes Kyon, but tends to squash her own feelings. Here though it's clear to everyone how Yuki feels, so Haruhi stands aside without a murmur and indeed helps her. Every so often, nonetheless, you'll see the most fleeting of hints. It's sad, although Haruhi would never ask for pity. Similarly we know that Koizumi's secretly in love with Haruhi, but knows she doesn't see him that way and so is never going to breathe a word about it. (You're more likely to think him gay from the evidence of this show alone, given his teasing of Kyon in ep.8, although being ambiguously bisexual is part of his root characterisation too.)
Both shows thus have faint, autumnal undertones of romance that can't happen, but in different ways. It's important, I think. Before, everyone had some degree of attraction towards Kyon, who in turn fancied Mikuru, but the only person allowed to get near Kyon was Haruhi and Kyon didn't like her. Now, it's Yuki-Kyon that trumps other possible relationships.
The two continuities overlap in other ways, too. I loved the Endless Eight references. The Remote Island Syndrome arc still happened offscreen, but with a bunch of strangers. It also pleased me when Kyon's little sister showed up, very late.
Then there's the eponymous Disappearance arc. It's impossible to talk about this in a spoiler-free way, although I'm itching to discuss it and could go on for pages. I love how they draw the lines to define the SPOILERS and never give us a temporary substitute of lesser status. Equal, but opposite. You can tell this was written by someone who loves the original characters. Oh, and I loved the hints of possible bleed, with a joke about murdering Kyon from someone who's already tried to kill him in multiple universes.
The differences are interesting. Kyon's less snarky, partly because he's no longer the narrator and partly because he's usually with Yuki (who doesn't deserve snark) rather than Haruhi (who does). Mikuru's been demoted to semi-regular and replaced by Asakura, who's the flatmate, best friend and pseudo-mother of the title character. Fair enough. As for Haruhi, she's still absolutely Haruhi, but no longer a jerk who makes you want to punch her in the face. She's likeable!
There's an OVA coming in October, by the way. I'll be looking out for it.
It's a quieter show. They're just students at school. It's gentle, to the point where a lot of viewers have been underwhelmed. If you're watching it just as a Yuki-Kyon romance, it's slow and uneventful. I could understand someone finding it dull. (The Disappearance arc is clearly stunning, though.) Personally, though, I thought the show was saying far more than just that. To me, those understated episodes feel rich and playful. Obviously I'd watched all of Haruhi quite recently and it was all fresh in my mind, but I enjoyed spending time with these people and seeing new sides of them. They felt like old friends, even though in the parent timeline I'm not sure I'd call them all that likeable. The finale resolves something fascinatingly eccentric and leaves more obvious things open, but I'm fine with that. Whether or not they make another season, I'm happy.
"I think you saved me from despair."