"It brings order to chaos."
A vinculum was a processing device at the heart of every Borg vessel which interconnected the minds of all the drones, purging individual thoughts and disseminating information relevant to the Collective. The vinculum was equipped with many safeguards to prevent its deactivation, and was capable of adapting to attack. It could also repair drones separated from the collective by bombarding them with personalities.
In 2375, Species 6339, a Delta Quadrant race which had been almost completely assimilated by the Borg, intentionally infected a Borg cube with a synthetic pathogen designed to attack the vinculum's programming as it would living cells. The drones aboard the cube subsequently began to experience an inability to suppress the various neural patterns of the individuals assimilated by the Borg, causing chaos to spread throughout the ship. Within three days of infection, the cube was destroyed. The vinculum, which survived the destruction of the ship, then established a connection with Seven of Nine's cortical implant via interlink frequency, causing her to develop the same symptoms as the crew of the cube. Seven began to experience extreme difficulty in controlling the identities which were surfacing, essentially developing the Borg equivalent of multiple personality disorder. The crew of the USS Voyager subsequently attempted to deactivate the vinculum by targeting a dampening field at its transneural matrix, but were unsuccessful in their attempts as the vinculum began adapting by rerouting its internal circuitry. Despite an attempt by Species 6339 to forcibly retrieve the vinculum from Voyager, Lieutenant Torres was successfully able to remodulate the dampening field to compensate for the adaptation. Captain Janeway then ordered the vinculum beamed into space, where Species 6339 retrieved it and withdrew from attacking Voyager. (VOY: "Infinite Regress")
The idea of the Borg vinculum originated with a story that was thought up and pitched by freelance writer Jimmy Diggs. The suggested plot involved the discovery of a destroyed Borg vessel. Executive producer Brannon Braga was interested in using the concept of the vinculum but initially insisted that the story featuring it would be, in the words of supervising producer Joe Menosky, "a maddening tech story." (Cinefantastique, Vol. 31, No. 11, p. 37)
Braga was given the challenge of incorporating the technicalities of the vinculum into the script for "Infinite Regress". "That vinculum ended up an albatross around [Braga's] neck," Joe Menosky recalled. "Brannon paid the price for it, because he had to do a massive amount of rewriting. It was a very difficult script to get in shape." (Cinefantastique, Vol. 31, No. 11, p. 37)
"Vinculum" is a Latin word meaning "a means of binding". Its literal translation is "tie", "chain", "bond", or "fetter". 
In the video game Star Trek: Voyager - Elite Force, set in 2376, Voyager is trapped in a region of dead space by a species known as the Vohrsoth (β). Among the other ships also stranded there is a Borg cube. After beaming over to the cube, the Hazard team make their way to the center of the vessel where they encounter the Vinculum. Once there, force fields engage around the team and the Vinculum possesses Seven of Nine who then offers them a deal - destroy an infestation of Species 8472 on the ship in exchange for a substance called isodesium (β) that could help Voyager escape. After dispatching Species 8472, the Borg attempt to assimilate the Hazard team but Ensign Austin Chang (β) detonates an explosive near the Vinculum, severely damaging it. In the chaos that follows, the Hazard team manages to escape the cube.
In Star Trek: Legacy, during the year 2380, a task force of vessels consisting of Federation, Klingon and Romulan ships is assembled to destroy the menace presented by a rogue Vulcan named T'Uerell (β), who is attempting to assimilate known space and impose her logic and order onto other races. After defeating the Borg under the command of T'Uerell, the task force targets and destroys her vessel. However, after leaving the battlefield, they fail to notice that the Vinculum of her ship managed to survive and is floating in space in a deactivated state. Once the fleet departs, the Vinculum becomes active and begins glowing with energy.
During the Star Trek Online mission "Assimilation", the player is dispatched to investigate Borg and Undine (β) activity. During the mission, the player beams aboard a Borg cube and is given the task of overloading it's vinculum before beaming back and making an escape.
In May 2017, Star Trek Timelines released a limited-time "event" series of missions called "Resistance", the storyline of which involved Starfleet attempting to destroy the Borg by studying the vinculum of a crashed Borg ship. Eventually, a pulse is developed capable of penetrating the shields of a vessel's vinculum, allowing Starfleet to silence every drone in the Collective.
The vinculum is also featured in the Star Trek Customizable Card Game which describes it's function is to, "Process and disseminate information relevant to the collective. Establish transneural connections between drones. Purge individuality; bring order to chaos."