The first Assassin's Creed game primarily takes place in 1191.
Its historical sections follow Altaïr Ibn-La'Ahad during the Third Crusade, when the Levant is contested by forces including the Crusader states and Saladin's armies. Altaïr travels among Masyaf, Jerusalem, Acre, and Damascus while investigating targets connected to the Templars and the Apple of Eden.
The year 1191 is part of the game's fictionalized historical framework rather than a claim that every event happened exactly as shown. The opening and closing sequences also occur in the modern era through Desmond Miles and the Animus, so the question's date refers to the main historical campaign. The First Crusade began in 1096 and captured Jerusalem in 1099, while 1291 marks the fall of Acre, a later endpoint often associated with the Crusader presence in the region.