The Barzan wormhole was a wormhole with one end, or terminus, initially located near the planet Barzan II in the Alpha Quadrant and one end that moved unpredictably to points through the galaxy, including the Gamma and Delta Quadrants, stabilized by intense energy fields within itself, until 2373 when its Alpha Quadrant end began to shift as well. (TNG: "The Price", VOY: "False Profits")
The wormhole became visible due to radiation buildup in the accretion disk, whose visible brightness was brief. When reaching the terminus, accretion matter could also be observed. Instability in the wormhole could be observed by increasing meson and lepton fluctuations as well as exponentially-increasing gravity fields.
The wormhole provided the Barzans with their first true natural resource, an opportunity which they decided to capitalize on by opening up negotiations for use of the wormhole in 2366. Present at the negotiations were representatives for the Federation, the Caldonians, the Chrysalians (represented by Devinoni Ral), and, later, the Ferengi.
During the time that the wormhole's Barzan location was stable, the wormhole became visible every 233 minutes. A Barzan probe sent through the wormhole discovered that the Gamma Quadrant terminus was located beyond the Denkiri Arm, a distance that at warp 9 would have taken nearly a century to cover.
Premier Bhavani of the Barzanian Planetary Republic hosted the bidding parties aboard the USS Enterprise-D. During the negotiations, Seth Mendoza represented the Federation's interests and presented their proposal, including: a lump sum payment of 1,500,000 Federation credits would be made upon conclusion of agreement, 100,000 credits per Barzanian year thereafter, the Barzanian Planetary Republic would be an equal partner in proceeds of operational revenues of venture, Barzanian personnel would be employed as principal operating staff of ground-based support facility, long-term economic, technical, and educational assistance would be provided by United Federation of Planets, and Barzan II would benefit from increased planetary security due to protection by Starfleet, with a potential agreement term that would have expired stardate 53000.
During the negotiations, the Enterprise conducted scientific studies of the wormhole, sending Lieutenant Commanders Data and Geordi La Forge through in a shuttlecraft, followed by two Ferengi, Arridor and Kol. Emerging from the wormhole, Data and La Forge discovered that, as opposed to arriving in the Gamma Quadrant as expected, they had emerged in sector 3556 of the Delta Quadrant, 200 light-years away. They quickly realized that although the terminus in the Alpha Quadrant was stable, the other end shifted positions periodically, and that, within a few centuries, both ends would eventually destabilize. Data and La Forge attempted to explain the situation to the Ferengi, who refused to accompany the Starfleet officers back through the wormhole. Data and La Forge returned to the Alpha Quadrant, but Arridor and Kol stayed, and they watched in shock as the wormhole destabilized, stranding them in the Delta Quadrant. (TNG: "The Price")
In 2373, the starship USS Voyager, which had become stranded in the Delta Quadrant because of the Caretaker's array, discovered Arridor and Kol living on a planet, in a region called Takar, posing as gods by using the religion of the local inhabitants to exploit them. The two lived lavish lives while the people were mostly in squalor. Captain Kathryn Janeway did not want to let the Ferengi continue with their gross exploitation of the inhabitants; at the same time, she and her crew tried to find a way to make the wormhole appear long enough for Voyager—and the two Ferengi they planned to take with them—to go through.
Voyager bombarded the area of the subspace instability created by the wormhole with verteron particles, which they hoped would amplify and polarize the instability, attracting the wormhole so they could travel through it back to the Alpha Quadrant. This was a risky undertaking, however; many different factors could deflect the wormhole from their sector, such as gravimetric shifts or solar flares.
Escaping Voyager in their shuttle, the two Ferengi initiated a high-energy graviton pulse, which changed the rotational momentum of the wormhole, destabilizing it. Although the gravitational eddy of the wormhole pulled the shuttle in, the graviton pulse had knocked the wormhole completely off its subspace axis, causing the wormhole to jump around erratically on both ends before disappearing altogether. This once again ended Voyager's hopeful attempts to return home. (VOY: "False Profits")
"The Price" was the first to use the term "Delta Quadrant" and established its extreme distance as well as the tendency for spatial phenomena located in it, such as wormholes, to be unstable.
According to the script, the pronunciation for Barzan was "BAR-zan". 
In the DS9 novel A Stitch in Time, Cardassians had suspected the existence of the wormhole nearly a decade before it was discovered, and Elim Garak studied it while he was at the Bamarren Institute for State Intelligence.