New Game+ Rings

Ring	Effect	Acquisition
Bloodbite Ring +1	Increases Bleed resistance by 190.	Found above the Smouldering Lake, next to the giant ballista.
Chloranthy Ring +1	Increases the Stamina recovery rate by 8 points/sec.	Found in Irithyll of the Boreal Valley, below the pedestal containing the Ring of the Sun's First Born.
Covetous Gold Serpent Ring +1	Increases Item Discovery by 75.	Found in Irithyll of the Boreal Valley, on a corpse on a ledge below a broken railing immediately after the dark room filled with Irithyllian Slaves, shortly before reaching the water.
Covetous Silver Serpent Ring +1	Fallen enemies will yield 20% more souls.	Found in Irithyll Dungeon, in the elevator shaft next to the first bonfire.
Dark Stoneplate Ring +1	Increases Dark damage absorption by 17.	Found in Lothric Castle, next to the white Lothric Wyvern. From the Dragonslayer Armour bonfire, turn back and stick to the right until you find an opening on the railing. Drop down and access a room. The ring will be on a corpse outside of this room.
Flame Stoneplate Ring +1	Increases Fire damage absorption by 17.	Found in the Profaned Capital, in an alcove on the roof of the front of building with three Monstrosities of Sin in the toxic water area.
Fleshbite Ring +1	Increases Poison, Bleed, Frost and Curse resistance by 60.	Found on the High Wall of Lothric, on a roof not far from the Tower on the Wall bonfire. Head out onto a large roof with one Pus of Man and go to the ladder where you will see the ring on the next roof over.
Havel's Ring +1	Increases maximum Equip Load by 16.5%.	Archdragon Peak treasure: From the Great Belfry bonfire, proceed down the stairs and make your way towards all the giant enemies on the left, they'll be patrolling near the only ruins you can see. The ring will be on a corpse in there.
Life Ring +1	Increases max HP by 6.5%.	Found in the Undead Settlement, on a ledge where Siegward of Catarina can be met, that is accessed by jumping off the tower elevator before it reaches the top.
Lingering Dragoncrest Ring +1	Extends the duration of Spell effects by 35%.	Found on the Road of Sacrifices near the Crucifixion Woods bonfire, behind the giant crab on a corpse.
Magic Stoneplate Ring +1	Increases Magic damage absorption by 17.	Found in Farron Keep, right from the Farron Keep bonfire, in the area filled with Basilisks.
Poisonbite Ring +1	Increases Poison resistance by 190.	Undead Settlement treasure: From the Dilapidated Bridge bonfire, head towards the cemetery area (where arrows rain down on you). The ring will be behind a well on a corpse.
Ring of Favor +1	Increases max HP by 4%, Stamina by 9% and max Equip Load by 6%.	Found in Irithyll of the Boreal Valley, above the Pontiff Sulyvahn bonfire.
Ring of Steel Protection +1	Increasesal Physical damage absorption by 13% against enemies. (3% against players)	Found in the Untended Graves, behind the tower that is unlocked with the Tower Key.
Ring of the Evil Eye +1	Restores 33 HP per enemy killed.	Found in the Cathedral of the Deep, behind the large structure right before the Deacons of the Deep arena.
Sage Ring +1	Increases Spell casting speed by ???% (Same effect to spell casting as +35 Dex)	Found in the Grand Archives, on the rafters below the top ceiling. (Can be reached by dropping down from the center of the Grand Archives' roof, where the Golden Winged Knights trio is encountered.)
Speckled Stoneplate Ring +1	Increases Magic, Fire, Lightning and Dark damage absorption by 7.	Found in the Cemetery of Ash, near the Ravenous Crystal Lizard.
Thunder Stoneplate Ring +1	Increases Lightning damage absorption by 17.	Found in the Catacombs of Carthus, in an area with many breakable pots just before the first bonfire.
Wolf Ring +1	Increases Poise by 6.	Found on the Road of Sacrifices, next to the Keep Ruins bonfire.
Wood Grain Ring +1	Slows equipment degradation by ???%.	Found in the High Wall of Lothric. From the Dancer of the Boreal Valley bonfire, climb the ladder and head left. Continue to the lift and go down. The ring is behind the lift at the bottom.

Additionally, there is one +2 ring that can be found on NG+:

Ring	Effect	Acquisition
Life Ring +2	Increases max HP by 7.5%.	Found in Lothric Castle. From the Dragon Barracks bonfire, cross the bridge and enter the room on the right. Climb a ladder and then exit through the door on the left. Once outside, head around to the right and look for the ring on a ledge below.