Anri of Astora is a fellow hollow trying to get the Lords of Cinder back to their thrones, and will be whichever gender your character isn't, but aside from being one of the only friendly faces you'll see, Anri is also part of a questline that will give you a different ending to the game if you follow it in the right direction.
To get this option you also need to have found Yoel of Londor, have drawn out your strength and hollowed enough that he dies and Yuria of Londor shows up, and should never cure your hollowing or remove your Dark Sigil. Between meeting up with Anri you should go back and talk to Yuria every so often, to make sure her half of the questline is triggering.
Anri, and their companion Horace the Hushed, are hanging around at the Halfway Fortress bonfire on the Road of Sacrifices. This is the first place you'll meet them, and if you speak to Anri here you'll get a brief overview of what they and Horace are planning to do next. You should then go to the Cathedral of the Deep and kill the Deacons of the Deep, and do this before you go into the Catacombs of Carthus. After you've killed the Deacons go back to Firelink Shrine and Anri and Horace will be standing on the stairs to the left of the empty thrones, and Anri will say that they now plan to go through the Catacombs to get to Irithyll.
Anri can be found twice in the Catacombs of Carthas. The first time they'll be upset they've been separated from Horace, and the second they will be further on in the Catacombs, still Horace-less. You can choose to go and find Horace or not (he's found his way to the Smouldering Lake) but he's sadly a mindless hollow if you do, and you should kill him rather than run away, or it may disrupt the questline. If you do kill Horace, never find him, or don't tell Anri that you have, then Anri should be at Irithyll of the Boreal Valley like they're supposed to be.
You will next find Anri, now permanently without a Horace, at the Church of Yorshka bonfire, and you can advance Anri's questline in two ways. In the corner of the church, right of the main doors if you stand looking at them from the inside, is a cluster of statues. Go up and hit them, because one of them is a pilgrim in disguise, and they're there to assassinate Anri. If you kill the assassin Anri will live, and will put down a summon sign in front of the Cathedral at Anor Londo to ask for your help fighting Aldrich, Devourer of Gods. If you do help then Anri will leave their sword for you with Ludleth of Corland at Firelink Shrine (this will only kill Aldrich in Anri's world, not yours, which raises a lot of questions about who is actually killing what and how many versions of the Dark Souls world there are, and which one is the real one).
Killing the assassin means that Yuria of Londor will now be in a huff with you and won't speak to you, closing off the Usurpation of Fire ending to the game.
If you let the assassin live then when you return to the church Anri will be gone, and if you speak to Yuria at Firelink Shrine after you've defeated Pontiff Sulyvahn she will inform you that your spouse is now ready for your wedding.
The Darkmoon Tomb is behind an illusory wall, naturally. The easiest way to find it is to start at the Anor Londo bonfire and spin the tower back down so it's connected to Irithyll again. At the bottom of the tower is a circular room, and inside it is a statue, which you can hit to find the stairs down to the tomb.
It's here that you talk to another pilgrim from Londor, and it's revealed that marrying Anri actually means stabbing Anri's corpse through the face very solemnly. With this done, though, when you touch the flame at the end of the game you'll take it for yourself and get a slightly creepy cutscene at the end, where you're surrounded by a lot of chanting pilgrims and looking all ominous and stuff.
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