Dark Souls 3 was a big step forward for the Souls franchise in terms of player expression. The possibilities for special builds and unique character development were immense because of the overhaul of items and equipment. Rings and weapons are only 2 examples of how players could mix and match their arsenals to do pretty much whatever they wanted.
Dark Souls 3 made huge changes concerning combining rings with the special effects of certain weapons is key to a lot of PvP builds, as well as defeating certain bosses. The whole Dark Souls 3 experience can change with the addition of a certain ring to a build so it's important to explore and experiment with what's already in the inventory.

Ring Of The Evil Eye And Greatswords

Greatsword build in Dark Souls 3
The Ring of The Evil Eye is a niche ring that grants its user HP when they slay an enemy. During regular boss fights, this wouldn't be useful because there's no way to gain HP since there's a single boss, and the HP doesn't heal on individual strikes, just kills. The best way to maximize on this ring is to use it during horde battles with lots of enemies around. With this, every kill of the horde pays the player back with some HP and makes the fight a little easier.
Greatswords make a fantastic combination with this ring because of the wide range of their attacks. With sweeping swings, enemies all around the player take great damage, and die, feeding the player HP as they fight. This combination is powerful but only useful in specific situations like the 
Deacons of The Deep boss battle.

Pontiff's left eye and The Dancer's Enchanted Swords

Dark souls 3 Dancer's blades
This combo is another niche pick for specific builds, but still pretty useful for a unique experience. Dexterity weapons in Dark Souls 3 are built to attack quickly and allow their user to stay mobile, not to do a lot of damage. Overall this tradeoff isn't too beneficial to the player, and they're better off using a weapon that yields more damage in single hits and shorter combos. This combination of ring and weapons changes the game entirely.
With Pontiff's left eye, players gain health for each consecutive hit, which means longer combos grant more health. Pairing this with the Dancer's Enchanted Swords which have the highest potential for consecutive hits due to their special stance, and the healing is actually substantial. This strategy works like something straight out of Bloodborne and is a refreshing change to Dark Souls 3.

Sage Ring and Witchtree Branch

Dark Souls 3 Witchtrer branch
The Sage Ring and Witchtree Branch work in tandem as a spell casting combo. The ring boosts the speed at which spells can be cast, and the catalyst does the same. On top of boosting casting speed, the Witchtree Branch has a special skill that buffs spells as well.
This combination leads to extra strong spells that come out way quicker than normal. The potential of the branch itself is nothing to write home about as a catalyst so this combination is more of a gimmick than anything unless further optimized with equipment and another catalyst, but it's a great place to start.

Hawk Ring and Dragonrider Bow

Drawing a bow in Dark Souls 3
For players trying to live out their dreams of playing a sniper in Dark Souls 3 , the Hawk Ring and Dragonrider Bow are a must-try duo. The Hawk Ring improves range and damage of projectiles, including arrows shot from a bow. The Dragonrider bow is the most powerful bow with the furthest range in the game so together, they make for the strongest bow build.
This build isn't great for trying to progress through the game but it's awesome if players want to cause chaos as an invader . Players have used this build to hide out in areas where they know they have the high ground, and shoot down at unsuspecting victims. What's great about this is that unless the targets have specific builds or armor, they will probably be killed in one hit from the Dragonrider Bow if optimized.

Farron Ring And Handmaid's Dagger

Closeup of a handmaid's dagger in Dark Souls 3
There're tons of unique skills and spells in Dark Souls 3 that help players diversify their playstyle and stand out among others, making the experience their own. FP is used to execute these skills and spells, and it's got limited uses as indicated by the FP bar in the HUD. There're items to increase FP and replenish it like the Ashen Estus Flask, but this duo does that work for you.
Both the Farron Ring and Handmaid's dagger can recover FP with successful hits, which goes a long way for players who rely on skills heavily. Since the dagger is so light and good for dealing critical hits, players can equip it as a secondary item without much burden.

Chloranthy Ring and Greathshield Of Glory

Dark Souls 3 Greatshield of Glory build
The Chloranthy Ring is a staple of Dark Souls game play that has been around since the first Dark Souls game. The ring speeds up stamina recovery which is great for any build in the game frankly. This powerful ring has no downsides to it so players should use it if they find themselves struggling to manage stamina.
Couple this ring with the Greatshield of Glory, and stamina recovery soars through the roof. The reason these two work so well together is that greatshields suffer very little chip damage from blocking and also the least amount of stamina depletion for blocking. Combining these two items makes it easy to block almost any hit without a guard break since stamina recovers too quickly to deplete at a meaningful rate.

Hornet Ring and Daggers

Dark Souls 3 dragon's peak gameplay
This combination is another classic found in any Dark Souls game for PvP. The Hornet Ring increases damage from critical hits, and daggers have the highest damage multiplier for critical hits in the game. Using these two together, damage for parries, backstabs, and guard breaks all hit numbers otherwise impossible.
Equipping a dagger as a secondary and a heavy weapon as the primary makes for high-pressure situations where the opponent becomes scared to block or attack recklessly for fear of being critically hit. If the stats are managed correctly, players using this strategy will usually kill in one critical hit, so it's worth investing the time into.

Stoneplate Rings And Dragon Tooth

Dark Souls 3 Dragon Tooth PvP
The red and blue Stoneplate rings grant fire and magic resistance respectively, which is useful for certain areas of the game and in PvP. What makes this work with Dragon Tooth is that the weapon also grants a big bonus to the damage absorption of fire and magic. Since Dragon Tooth is so heavy, it's usually assumed that players will be donning heavy armor with their equipment.
Heavy armor decreases physical damage heavily but can lack in magic resistance. Combining these rings with this weapon together in battle creates a heavy tank capable of taking a lot of hits and dealing massive damage.

Ring of Favor And Heavy Weapons

Fighting knights in Dark Souls 3
The Ring Of Favor made its first appearance in the original Dark Souls and still has plenty of uses in Dark Souls 3 for PvE purposes. The ring buffs HP, stamina, and equip load considerably, granting the player abilities to do things they couldn't do without it. These stats together are great for using heavy weapons that normally wouldn't be effective due to insufficient stats.
RELATED:
Dark Souls The 5 Best Armor Sets In The Game (& The 5 Worst)
Heavy weapons are generally solid for PvE purposes because of their wide range and big damage that destroys mob enemies and midbosses. This isn't a specific build or setup, but it's good to know that this option exists for players that want to try out heavy weapons without committing to leveling up the stats first.

Tearstone Ring And Morion Blade

Dual Wielding Morion Blades in Dark souls 3
The Tearstone Rings of Dark Souls are one of the most iconic staples in the game for speed runs and challenge runs . They enhance the stats of the player as long as their health is below a certain threshold. This can put a handicap on the player since they can't take as much damage as usual or heal if they want the bonus, but the stat boost is worth it in certain situations.
Combining the Morion Blade with the Red Tearstone Ring doubles the effect of strengthening attack when HP is low. Players use this build to style on opponents in PvP and get easy one=hit kills that the opponent rarely can see coming.