"Bari" comes from "baria", the Japanese pronunciation of the word "barrier".
In the Viz Manga and FUNimation subs, it is called the Barrier-Barrier Fruit.
Strengths and Weaknesses
The main strength of this fruit is that it allows the user to form invisible barriers, though they can be solidified to become visibly transparent, resembling a pane of glass. This protects the user from attacks from either front or back. The barriers have immense defensive power, able to defend against even a fish-man's punch that can shatter 4000 tiles, which injured said fish-man's hand in return.  It also successfuly defended against Hakuba's attacks and the shockwave produced from Elizabello II's King punch, which could destroy an entire fortress. Bartolomeo has a high estimation of his barriers' strength, as he was confident that they were "unbreakable" and could stop the Birdcage (which can slice buildings and meteorites with great ease).Kurozumi Semimaru demonstrates that the barrier can also withstand an extremely powerful slash from someone as powerful as Kozuki Oden, a master swordman, whose sword could cut the Mountain God.
The shield can also be used for offensive purposes by launching it forward to create powerful forces to strike the targets with, or to deflect forces sent by enemies right back at them. According to Bartolomeo, he can create his barriers by crossing his fingers.
Another strength of this fruit is that Bartolomeo can form solid constructs out of his barriers, such as stairs which people can stand on, making this fruit useful for getting to hard-to-reach places.
According to Oda, the barriers created can cover a range of 50,000 Bari Baris.
It is unknown if the barriers have a damage limit, but they still have weaknesses. Bartolomeo has stated that the barriers are not soundproof. An additional limitation is that he can only sustain one barrier at a time, and that there is a maximum total surface area that the one barrier can cover; examples of the hazardousness of this weakness include being unable to completely block Pica's giant stone fist, as well as being incapable of producing more barriers after crafting a staircase out of them. The barriers are also not entirely immovable by outside forces, as seen when Bartolomeo tried to block the Birdcage. Also, certain barrier shapes could still be outflanked by going around, above, or below it especially if the user is ambushed or attacked by an enemy with superhuman speed. However, this normally isn't an issue because Bartolomeo can freely morph the shape of his barrier even after it is deployed. Bartolomeo does need to move or alter his barriers in order to attack, trading the barrier's defensive protection for offense. The fruit does not give the user superhuman durability, meaning that they can be vulnerable if the barrier is bypassed. Besides these weaknesses, it shares the standard weaknesses of Devil Fruits.
Semimaru, the former user of this fruit, used its power to protect Wano shogun Orochi from being killed by a furious Kozuki Oden.
Bartolomeo uses the fruit's power to protect himself in combat, sitting idly by while his unsuspecting foes attack him and injure themselves. He can also use the shields offensively to strike down and repel his targets, or even as walls to immobilize and crush his victims.
The barriers can also be used for supplementary purposes, such as a foothold, and Bartolomeo can form them into stairways to reach higher areas.
Barrier (バリア,Baria?): Bartolomeo activates his fruit's ability by crossing his fingers, causing a transparent barrier to appear which can protect him from attacks. So far, two shapes for his barrier have been seen, a rectangular barrier and a semi-spherical barrier.
Barrier Crash (バリアクラッシュ,Baria Kurasshu?): Bartolomeo forms a transparent barrier in front of himself and launches it towards the enemy at high speed, smashing it against them. The attack is powerful enough to defeat Elizabello II and create a shock wave. It was first shown being used against Elizabello in Corrida Colosseum right after it shielded Bartolomeo from the King Punch.
Barrier Bulls (バリア突進牛,Baria Buruzu?, literally meaning "Barrier Charging Bull"): Bartolomeo forms a curved transparent rectangular barrier in front of himself and charges forward as he pushes the barrier along, knocking and mowing everything in his path like a bulldozer. First shown against Gladius and some of Doflamingo's soldiers, as well as Cavendish who happened to be in the way.
Barrierbility (流動防壁,Bariabiriti?, literally meaning "Flux Barrier"): Bartolomeo activates this fruit's ability by squeezing his hands while crossing his fingers to change the shape of his barrier into various forms while retaining the barriers' hardness.
Stairs (階段,Suteāzu?): He changes the shape of his barriers into a flight of stairs, allowing him or his allies to move freely in the air while protecting them from attacks coming from below. It was first shown being used to help Luffy and Law reach the fourth level of the King's Plateau.
Racket (ラケット,Raketto?): He changes the shape of his barriers into a large tennis racket. He can extend this barrier to protect somebody or something with this ability. It was first shown being used to protect Nico Robin from Gladius' Catapult Punc attack.
Barrier Ball (バリアボール,Baria Bōru?): Bartolomeo creates a spherical barrier, preventing whatever is inside from escaping and whatever is outside from penetrating. It was first shown being used in an attempt to contain Gladius' attacks.
Homage Shinken: Bari Bari no Pistol (オマージュ神拳 バリバリの銃,Omāju Shinken: Bari Bari no Pisutoru?, literally meaning "Homage God's Fist: Bari Bari Pistol"): Bartolomeo forms a small spherical barrier around his fist and then punches an opponent, using the barrier to amplify the strength of the punch. It was first seen being used to finish off Gladius. It is an obvious and intentional homage to Luffy's Gomu Gomu no Pistol. It is called Barri-Barri Pistol in the english localization of One Piece: Burning Blood.
Big Hand (,Big Hand?): Using Barrierbility, Bartolomeo shapes a barrier into a giant hand and pushes it forward, allowing him to contain and push things. It was first used against Bill to push away his massive flow of molten ore.
Anime and Manga Differences
In the anime, Bartolomeo claims he is unable to deploy barriers if he cannot cross his fingers, contradicting to when he encased his fist to strike down Gladius. The anime addressed this by having Bartolomeo cross his fingers with his other hand.
Contradicting this again in the anime Bartolomeo uses the technique again during a montage in Episode 745, where he is not seen to cross his fingers in generating the attack, and his left fist can be seen visibly clenched immediately before and after it is generated (apart from the single frame when it is generated where it would not have been possible for him to unclench and reclench his fist).
It can be considered ironic that Bartolomeo has this Devil Fruit because while his behavior is generally provocative, this Devil Fruit is defensive by nature.
The sign Bartolomeo performs with his fingers when summoning his barrier is commonly known in Japan as a gesture to either to repel "bad things", or provoke others by saying they are being repelled as they said "bad things". The gesture is a common taunt for school-age children and delinquents.