Anesthizine is a fast-acting anesthetic, administered through environmental systems as a gas. It is considered among "the best", along with axenol and neurozine, during the late 24th century. (VOY: "Message in a Bottle") A concentration of 70 ppm is typically enough to put a humanoid to sleep. (TNG: "The Hunted")
Uses
While attemepting to subdue Roga Danar in 2366, Jean-Luc Picard ordered that all the cargo bays on Deck 38 be flooded with anastazine. After flooding the bays with the gas, it was realized that Roga had escaped the effects of the gas by exiting the bay via use of a pressure suit. (TNG: "The Hunted")
While attempting to incapacitate the three alien-possessed USS Enterprise-D crew that was holding Ten Forward hostage in 2368, Riker suggested flooding the air vents with anesthizine gas. La Forge, however, turned the idea down, noting that the gas wouldn't affect Data. (TNG: "Power Play")
In 2369, when the Ferengi commandeered the USS Enterprise-D, Ro Laren suggested that they could flood the ship with anesthizine gas, as an option to retake the ship. Jean-Luc Picard turned the option down, however, noting that access to the particular system needed to administer the gas was locked out along with the ships other command functions. (TNG: "Rascals")
The besieging General Krim, of the Bajoran Militia, attempted to flood the conduits of Deep Space 9 with anesthizine gas, starting with level five, in attempt to flush out the insurgents, lead by Benjamin Sisko, who were stowed away on the station following its evacuation in 2370. (DS9: "The Siege")
When the USS Prometheus was commandeered by the Romulans in 2374, the Doctor attempted to devise a means that they could incapacitate their captors by the use of anesthetics. Offered the options of axenol, neurozine, anesthazine, by the EMH Mark II, the Doctor ultimately chose neurozine as the best choice. (VOY: "Message in a Bottle")
Later that year, Sisko flooding all compartments of the USS Defiant, except the engine room, with anesthezine gas after the crew managed to commandeer engineering from the Jem'Hadar. (DS9: "One Little Ship"
Background
At least six different spellings have been used by the writers in desribing this gas in various scripts. In "The Hunted", it was spelled as both anastazine and anestazine; in "Power Play", it was spelling anestizine, in "Rascals", it was spelled anesthizine; in "The Siege", it was spelled as anesthizine; in "One Little Ship", it was spelled anesthezine; and in "Message in a Bottle", it was spelled anesthazine.
Anesthizine is the spelling choice used here because it was the spelling that was used the most-- twice, and is also the choice of spelling used by the official website.