Atrisian scribes wrote in that script the Sayings of Uueg Tching who taught his heirs, and Uueg intended them to be a guide to his descendants.
At the time of the Cold War, many inscriptions and advertisements in Atrisian script could be spotted, notably on Nar Shaddaa. This means this script was already in wide galactic use as of 3653 BBY.
Millennia later, by the time of the Galactic Civil War, it was most commonly used for confidential information.
The Galactic Empire made use of in Imperial installations where one entered a combination using letters from the Atrisian alphabet to access some sections; it was Officers who typically carried that showed each 3-letter combination. Some crates stored in Ramsees Hed had labels with Atrisian characters.
The "Imperial Clock" screensaver in Star Wars Screen Entertainment
It seems the (then unnamed) Atrisian script was first used in old LucasArts games before Stephen Crane had fully developed Aurebesh. It appears to be the same as the writing on the display in the Emperor's throne room aboard the Death Star II (seen in Star Wars: Episode VI Return of the Jedi).
An ornate version of this alphabet appeared in 1994's Star Wars Screen Entertainment, in a module that was a re-creation of the Death Star Viewport targeting Yavin 4.