This achievement requires collecting all 107 rings, which includes all variations. Therefore you need all 70 basic rings AND all +1 and +2 versions (one ring has a +3 version, too).

The basic versions of each ring can be found in the first playthrough. The +1 versions can be found in playthrough 2 (in unique locations). The +2 versions (and single +3 version) can be found in playthrough 3 (again, in unique locations). In subsequent playthroughs, all of the rings from previous playthroughs are available as well, so you can't miss out on anything that you didn't get by going to the next playthrough.

Most are found in static places throughout the world. Some are sold by vendors. Some are are only available as covenant rewards and can't be obtained otherwise.

Rings can be traded between players, so if you have a friend that has a ring that you're missing and they drop it for you, it will count towards the achievement.

Ring of Life
Starting gift or sold by the Handamaid in Firelink Shrine after returning the Dreamchaser's Ashes. These can be found on a corpse behind an illusory wall near the Old Wolf of Farron bonfire.

Ring of Life +1
On a Ledge where you meet Siegward in the Undead Settlement that is accessed by jumping off the lift before it reaches the top.

Ring of Life +2
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. Can be found in NG+.

Ring of Life +3
Behind the large throne in Dark Firelink in the Untended Graves.

Sun Princess Ring
Found in Gwynevere's Chamber above the Aldrich, Devourer of Gods bonfire.

Estus Ring
Found behind the locked door on the left side of Firelink Shrine accessible by going through the locked door on the right side of the area (which requires the Tower Key from the Handmaid to unlock) and dropping down via coffins sticking out of the wall.

Ashen Estus Ring
Proceed from the Untended Graves bonfire and take left path. The ring is on a corpse behind some crow enemies.

Chloranthy Ring
Under breakable crates/barrels in the bottom of the tower that you go through to exit the Siegeward/Fire Demon area in Undead Settlement.

Chloranthy Ring +1
From the Pontiff Sulyvahn bonfire, backtrack through the initial entrance and drop off a ledge straight ahead to land on a structure below. The ring in on ground level behind this structure.

Chloranthy Ring +2
Drop down to a ledge to the left of the first Crow Mage in Road of Sacrifices.

Havel's Ring
Trade with Ludleth of Courland for the Soul of a Stray Demon.

Havel's Ring +1
From the Great Belfry bonfire in Archdragon Peak, head outside and follow the path towards the altar. It's on the left side in a corner just through the second stone arch.

Havel's Ring +2
On a rafter as you work your way down the tower near the Prison Tower bonfire.

Ring of Favor
Obtained by killing the two huge dogs by the Aldritch Faithful NPC in the area beyond he Pontiff Sulyvahn bonfire.

Ring of Favor +1
In a walkway above the Pontiff Sulyvahn bonfire.

Ring of Favor +2
From the Cleansing Chapel bonfire, head through the door on the left and take a lift up to a tower. Go around the outside of the tower and climb a ladder to the top and then drop down to a slanted platform that leads up to a roof. Turn left and an axe wielding enemy will come out of an alcove. The ring is in this alcove.

Prisoner's Chain
Trade with Ludleth of Courland for the Soul of Champion Gundyr.

Ring of Steel Protection
In the Ancient Wyvern area, take the right path to the end and go around a railing.

Ring of Steel Protection +1
At dark Firelink Shrine in Untended Graves behind the tower on the far right.

Ring of Steel Protection +2
Visible on a platform in the Catacombs of Carthus accessed by dropping off the left side of the first long bridge.

Magic Stoneplate Ring
Dropped by one of the two large knights outside Oceiros, the Consumed King's fog door.

Magic Stoneplate Ring +1
Head to the right from the Farron Keep bonfire to an area with cursed basilisk enemies. It's on a corpse in this area.

Magic Stoneplate Ring +2
From the Profaned Capital bonfire, climb down the interior ladder, exit through the hole in the wall, and turn right. The ring will be towards the end of this path.

Flame Stoneplate Ring
On a corpse hanging from a tree in the Undead Settlement by the Mound Makers cage guy. It may drop all the way down to in front of the building with the lift that leads to the giant archer.

Flame Stoneplate Ring +1
Near the Profaned Capital bonfire, in an alcove on the roof of the front of building with the three big rhino/hand guys in the toxic water area.

Flame Stoneplate Ring +2
In a room behind an illusory wall in the Demon Ruins area below Smouldering Lake where a Black Knight is fighting another enemy.

Thunder Stoneplate Ring
Proceed from the Dragon-Kin Mausoleum bonfire until you reach an outside area with some scaffolding. Climb a ladder in this area to reach the platform where the ring is located.

Thunder Stoneplate Ring +1
In an area with many breakable pots just before the Catacombs of Carthus bonfire.

Thunder Stoneplate Ring +2
Backtrack from Dragonslayer Armor bonfire and go left into the building with a lift. Go through an opening on the left to get outside on a ledge. Drop down from that ledge onto a roof and the ring is on the right side.

Dark Stoneplate Ring
In the lower area to the left of the open area with giants just past the Pontiff Sulyvahn bonfire (below the area with several mags enemies).

Dark Stoneplate Ring +1
Backtrack from the Dragonslayer Armor bonfire and drop through a break in the railing on the right side of the area to access a room. The ring is on a ledge outside of this room.

Dark Stoneplate Ring +2
From the Keep Ruins bonfire, look behind the building beyond the three enemies on the bridge.

Speckled Stoneplate Ring
Behind a breakable brick wall in the Smouldering Lake area of the Catacombs of Carthus (must be broken by ballista fire).

Speckled Stoneplate Ring +1
From where you start each playthrough, proceed straight ahead and then to the right to an area with a crystal enemy. The ring is in this area.

Bloodbite Ring
Obtained by killing the large rat in the area across the bridge from the Dilapidated Bridge bonfire in the Undead Settlement.

Bloodbite Ring +1
Near the giant ballista above Smouldering Lake deep in the Catacombs of Carthus.

Poisonbite Ring
From the cleansing Chapel bonfire, head outside and turn right from where the four enemies are praying. Drop down and follow the path around until you reach an area with a large crystal enemy among some blue crystals. There will be an entrance to a grey stone area nearby where the ring is inside on a corpse.

Poisonbite Ring +1
Behind a well in the cemetery area of the Undead Settlement.

Cursebite Ring
From the Profaned Capital bonfire, work your way to the area with the toxic swamp. The ring is on a corpse in the cave that you can drop into via the building above.

Fleshbite Ring
In the Grand Archives, cross the bridge and proceed past the shortcut ladder by four Mage enemies and some bookshelves. The ring is on a corpse one level up in an area with a bunch of books and cursed hands.

Fleshbite Ring +1
On a roof not far from the Tower on the Wall bonfire. Head out to the roof area with one of the transforming enemies and go towards the ladder and you will see the ring on the next roof over. You must run and jump across to get there.

Knight's Ring
In Lothric Caste in a room before the Sunlight Altar.

Hunter's Ring
Climb high onto the roof of the Grand Archives where three flying knights engage you. The ring is on a corpse at the top of the central tower in this area.

Scholar's Ring
You can pull a lever next to the bridge in the center of the Grand Archives to lower an adjacent bookshelf revealing a corpse with the ring.

Priestess Ring
Sold by the Handmaid in Dark Firelink Shrine in the Untended Graves.

Covetous Gold Serpent Ring
In the locked cell with Siegeward in Ithiryll Dungeons. The key to the cell is in the right side chest at the end of a sewer hallway leading away from the area with the giant.

Covetous Gold Serpent Ring +1
From the Church of Yorshka bonfire, proceed into a graveyard and the head left and into a dark room full of shadow enemies. Immediately upon exiting this room, drop off the right side to a ledge to find the ring.

Covetous Gold Serpent Ring +2
On a corpse in the area accessible after defeating the Nameless King.

Covetous Silver Serpent Ring
Behind an illusory wall above the Firelink Shrine bonfire. You need the Tower Key from Handmaid to access this area.

Covetous Silver Serpent Ring +1
Take the shortcut door next to the Ithiryll Dungeon bonfire to get to a lift. Jump off early to reach a platform where the ring is on a corpse.

Covetous Silver Serpent Ring +2
Visible on a ledge just below Flynn's Ring in the Siegeward/Fire Demon area of the Undead Settlement.

Saint's Ring
Sold by Irina of Carim in Firelink.

Deep Ring
Dropped by an enemy in a tower accessible by a lift near the Cleansing Chapel bonfire.

Darkmoon Ring
Reward for reaching rank 1 in the Blade of the Darkmoon covenant. The Silver Knights in Anor Londo can drop Proofs of a Concord Kept (very rare), and you need 10 to reach rank 1.

Young Dragon Ring
Given by Orbeck of Vinheim after returning scrolls and buying sorceries.
Note: He gave this ring to me after returning two scrolls and buying five or so spells, then talking to him.

Bellowing Dragoncrest Ring
From the Irithyll Dungeon bonfire, proceed into the large room and follow the hallway to the right to a locked door. Open this door with the key from a cell upstairs on the other side of the large room and drop to a platform where this ring is on a corpse.

Great Swamp Ring
Dropped by a large crab in Crucifixion Woods.

Witch's Ring
On an altar at the end of the hallway just before the Abandoned Tomb bonfire in the Catacombs of Carthus.

Morne's Ring
Backtrack from the Halfway Keep bonfire in Crucifixion Woods to a bridge. You can drop to a ledge below and the ring is on a corpse nearby.

Ring of the Sun's Firstborn
Backtrack from the Pontiff Sulyvahn bonfire and fall off the ledge to a platform below.

Lingering Dragoncrest Ring
Obtained upon killing a giant crab in Farron Keep near the Keep Ruins bonfire.

Lingering Dragoncrest Ring +1
On a corpse in the watery area of Crucifixion Woods near the giant crabs.

Lingering Dragoncrest Ring +2
Climb high onto the roof of the Grand Archives where three flying knights engage you. The ring is in an alcove on the roof immediately above the area where the knights land.

Sage Ring
From the Crucifixion Woods bonfire, drop into the water and follow along the wall on the left until you see an opening. The ring is on a corpse in this area.

Sage Ring +1
Climb out onto the roof of the Grand Archives where three flying knights engage you. High atop this roof, you can drop into the inside of the building and then work your way down some rafters. The ring is on a corpse on one of the rafters.

Sage Ring +2
From the Dancer of the Boreal Valley bonfire, climb the ladder and head left. Once outside, follow the path to a lift and jump off halfway down. Head left and fall onto a platform, then jump to an adjacent platform where the ring is.

Dusk Crown Ring
On a corpse in a cell in the lowest level of Irithyll Dungeon.

Leo Ring
In a chest on the second level of the large room just beyond the kitchen area in Ithiryll of the Boreal Valley somehwat close to the Distant Manor bonfire.

Wolf Ring
Obtained by reaching rank 2 in Watchdogs of Farron covenant. This requires 30 Wolf's Blood Swordgrass.

Wolf Ring +1
From the Keep Ruins bonfire, exit and circle the building to find this ring on a corpse.

Wolf Ring +2
On the left immediately outside the Iudex Gundyr area.

Hawk Ring
Dropped by the giant archer atop the tower in the Undead Settlement.

Hornet Ring
On a path to the right of Dark Firelink shrine in the Untended Graves.

Knight Slayer's Ring
Obtained by killing Knight Slayer Tsorig as either a red phantom invader near the bridge in the Catacombs of Carthus or as a human enemy in a lava area below Smouldering Lake.

Ring of the Evil Eye
Given by Anri at his/her second location in the Catacombs of Carthus (or at the Church of Yorshka). You must first talk to him/her at the Halfway Fortress bonfire, then at Firelink Shrine, then in two places in the Catacombs of Carthus.

Ring of the Evil Eye +1
Behind the large structure right before the Deacons of the Deep boss room.

Ring of the Evil Eye +2
In a corner behind some barrels in the room in High Wall of Lothric where Greirat's cell key and the first Estus Shard is found.

Farron Ring
Given by Hawkwoon in Firelink Shrine after defeating the Abyss Watchers.

Dragonscale Ring
Take the shortcut lift down to the toxic swamp before Oseiros, the Consumed King's room but jump off early to access the path where the ring is on a corpse.

Horsehoof Ring
Sold by Patches in Firelink Shrine.

Wood Grain Ring
Sold by the Handmaid in Firelink after returning the Eastern Ashes from the area just past the Pontiff Sulyvahn bonfire.

Wood Grain Ring +1
From the Dancer of the Boreal Valley bonfire, climb the stairs and head left. Continue to the lift and take it down to the Consumed King's Garden. The ring is behind the lift at the bottom.

Wood Grain Ring +2
In Ithiryll of the Boreal Valley, after exiting the building with a Silver Knight shooting you with a greatbow, there are stairs to the left but instead turn right and look around the corner for the ring.

Flynn's Ring
On the highest roof in the area where you fight the Fire Demon with Siegeward in the Undead Settlement.

Magic Clutch Ring
Behind an illusory wall near the Church of Yorshka bonfire (one level below the Pontiff Sulyvahn fog door on the left).

Lightning Clutch Ring
In an alcove to the left of the entrance to the Ancient Wyvern area in Archdragon Peak.

Fire Clutch Ring
In the Undead Settlement near the Cliff Underside bonfire. It's at the end of a wooden structure behind some gallows.

Dark Clutch Ring
In a mimic chest at the end of a sewer hallway that connects to the large lower level of Ithiryll Dungeon.

Red Tearstone Ring
Backtrack from the Dragonslayer Armor boss room and go into the building on the left (with a lift). Go outside through an opening in the wall and the ring is on a corpse.

Blue Tearstone Ring
Given by Greirat in High Wall of Lothric.

Lloyd's Sword Ring
On a corpse on the ledge that you run around to get past the first giant in the Cathedral of the Deep.

Lloyd's Shield Ring
Sold by the Handmaid in Firelink Shrine after the Paladin's Ashes are returned.

Carthus Milkring
In an area with many breakable pots just before the Catacombs of Carthus bonfire.

Carthus Bloodring
Head left from the Catacombs of Carthus bonfire, then take two rights and follow a path that leads to some wheel skeleton guys. The ring is on a corpse at the end of this path.

Pontiff's Right Eye
Obtained by killing the large dog enemy on the bridge upon entering Itithyll of the Boreal Valley

Pontiff's Left Eye
Trade with Ludleth of Courland for the Soul of Boreal Valley Vordt.

Aldrich's Ruby
Dropped by a large spider enemy in the large room in Anor Londo that spawns when you get close the mechanism to open the doors.

Aldrich's Sapphire
Dropped by a large spider enemy near the huge room with the two giants in the Cathedral of the Deep.

Silvercat Ring
Given by Sirris of the Sunless Realms in Firelink Shrine after using her summon sign on the Boreal Valley bridge and defeating the enemy.

Slumbering Dragoncrest Ring
Given by Orbeck of Vinheim after returning scrolls and buying sorceries.
NOTE: He gave this ring (and his gesture) to me after returning two scrolls and buying 15 or so spells, then talking to him.

Untrue Dark Ring
Sold by Yuriel of Londor.

Untrue White Ring
Sold by Yuriel of Londor.

Obscuring Ring
Reward for reaching rank 1 in Rosaria's Fingers. This requires 10 Pale Tongues.

Reversal Ring
In the back of the Darkmoon Tomb accessible right before entering Anor Londo.

Skull Ring
Appears on Ludleth's corpse in Firelink after killing the four Lords of Cinder, placing their remains on their thrones, and interacting with the bonfire.

Calamity Ring
Use the Path of the Dragon gesture by the altar right next to the Dragon-Kin Mausoleum bonfire in Archdragon Peak.

Ring of Sacrifice
Found on an ledge/awning above a courtyard in High Wall of Lothric.
Found in the watery area between the Church of Yorshka and Distant Manor Bonfires.
Sold by Yuria in Firelink.