Borodin was a chemist by trade (and also also a big agitator for women's rights, coolly enough); he just wrote music on the side as a hobby. He worked on this, his only opera (based on a medieval epic poem), on and off for eighteen years, leaving it unfinished at the time of his death: he'd written a lot of music (though some only in outline), but parts of it had no libretto, and parts weren't there at all. So his friend Nikolai Rimsky-Korsakov and his, Rimsky-Korsakov's, student Alexander Glazunov wrote extra music and pounded the whole thing into some semblance of order.
This Met production is very different from that. Director Dmitri Tcherniakov radically reworked the material: most or all of Rimsky-Korsakov's and Glazunov's music has been stripped away and music by Borodin that they apparently didn't have access to has been added back in, along with stuff that he wrote for other purposes (some of which he had cannibalized from the unfinished Prince Igor score--this gets very complicated). Furthermore, Tcherniakov changed the emphasis. One thing that frustrated Borodin and prevented him from completing the opera was a feeling that the story lacked dramatic movement, so now it's more of a psychological story about Igor's mindset (albeit not entirely convincingly), as he goes to war, is captured, and returns home to ruins.
So the plot: well, I guess that last sentence is more or less "the plot," such as it is. It's interesting in that usually you have a dichotomy: either an opera's a comedy or a tragedy. This really isn't either, though; it's certainly never funny or trying to be funny, but at the same time it has a happy or at least hopeful ending. Go figure.
For obvious reasons, Prince Igor is a very messy opera; there's no getting around that. I honestly was feeling sort of alienated at first. The opening scene, where the Russians are gonna go to war and Igor's great rah rah felt a bit pro forma, and then the second scene...my goodness. See, this is the point where Igor has been captured by the Polovtsians. And in this production--this is the most famous thing; you'll see it any promotional still--this takes place in a somewhat hallucinogenic field of poppies, indicating that this is taking place in part (but only in part--I really must insist on that) in Igor's mind. And at first it just felt very opaquely avant-garde to me, no clear idea on how anyone was feeling or what the purpose was or...huh. But I definitely perked up as things started to become a little clearer. This scene also features the famous "Polovtsian Dances," which are often performed in concert. And they're great.
The second act is a bit more "normal," returning home where Igor's wife is lamenting his absence and her vile brother is debauching it up. It's probably also the most accessible act, with some really great music especially as said brother, Prince Galitzky, fantasizes about his hedonistic would-be reign. Great chorus work, too: in my experience (sample size: 3) Russian opera tends to be big on the choruses. It ends as the Polovtsians attack. The third and final act (there's actually supposed to be an extra one after the second, but it's excised here and elsewhere for mostly having been written by Glazunov and for lacking forward momentum. It doesn't make a big difference, but its inclusion would've clarified a few things) is a bit less exciting; Igor, having escaped, is now home. And they begin to rebuild.
Boy, just getting your head around the plot there is kind of hard. I haven't even talked about the singers. Well, no one is less than "fine," as you'd expect. I think for me, the standouts are Mikhail Petrenko, as Galitzky, playing up the villain role to the hilt; Štefan Kocán as the Khan, who comes with real presence and a slightly fey affect that makes sense given the Orientalist conception of the Polovtsians; and of course Anita "great in everything" Rachvelishvili is predictably great as the Khan's daughter who falls in love with Igor's son. As Igor himself, Ildar Abdrazakov is basically fine, though he strikes me as just a bit too youthful and not quite forceful enough to fully inhabit the role.
There IS, it's hard to deny, some disturbing nationalist subtext here. When the Khan--who is treating Igor more as a guest than a prisoner--is talking with him in the poppy field, Igor's all oh but I don't have my FREEDOM and the Khan's like, okay, sure, just promise you won't attack me again and I'll let you go. But, predictably, he answers, nope, I am TOTES going to attack you again, and you think, good god, what is the PURPOSE of all this bloodshed? And...I mean, especially given Russia's current relations with some of its neighbors, it all seems extremely dubious.
I mean, you don't have to think about it in those terms. You probably shouldn't. There's plenty of great music here, so you should appreciate that, even if it has some slow bits and perhaps never quite comes together as a whole.