The opening sequence of the original Assassin's Creed is set in 1191.
The game's historical sections begin in 1191, during the Third Crusade. Altaïr Ibn-La'Ahad is the playable Assassin, and his missions take him through Masyaf, Jerusalem, Acre, and Damascus. These cities are presented through a fictional conflict between the Assassin Brotherhood and the Templars.
The year places the story within the period of the Third Crusade, which lasted from 1189 to 1192. The game includes historical figures such as Richard the Lionheart and Saladin, but its central conspiracy, characters, and supernatural elements are fictionalized for the series.
1191 should not be confused with 2007, the year the video game was released. Nor is it the year 1099, when the First Crusade captured Jerusalem, or 1453, the year Constantinople fell to the Ottoman Empire. The question concerns the in-game historical opening, not publication history.