The Chalice of Order is an artifact Connor obtained after unlocking a door with the Key of Order.
Background
It is said that the Chalice of Order provides the life immortal. Thus Connor may have gained immortality from the Chalice. It is also said he would someday return to his homeland but would never perish. Although this is unclear if that prophecy was just pointing out that he would survive his quest or if he would live forever.
Behind the scenes
The Chalice of Order is inspired by the Holy Grail. It is also known as the 'Grail' in the debug mode.
Quest for Daventry also makes reference to the Holy Grail as well, for whatever reason.
There is no specific use for the Chalice other than picking it up gives the 'precept of Order', needed to open the final talking door, the door of Order. However, there is a sound when its grabbed as if Connor drank from it.
The Chalice is also one of three items (including the Sword of Truth and Armor of Light) needed to enter the Inner Sanctum of the mask. However, the only way to find out about this is to use debug console cheats, and reach the Inner Sanctum level without picking collecting all three items. The Archon will mention to Connor that he must go back and get those treasures (as well as the fifth mask piece if he hasn't).