Harrowing

The Harrowing is a test that every mage Apprentice must go through to become a full member of the Circle of Magi. All of their training and apprenticeship is meant to prepare them for this test. It is practiced throughout Thedas, even as far as the Tevinter Imperium. Exactly what happens during the Harrowing is kept secret from apprentices, and from people outside the Circle.
The Harrowing looms large for apprentices, in part because it is the job of Enchanters to decide, on their own and in secret, when and if the apprentice is ready for the rite. When they do so, the apprentice is taken - without warning or preparation - to face the Harrowing. It can come at any time, and so one must always be ready.

Execution

The rite itself is fairly simple. First, the apprentice enters the Fade through ritual use of lyrium. Once across The Veil, the subject must face and overcome a Demon that wishes to possess the apprentice's body and enter the living realm. This involves navigating whatever challenges have been set up and seeing through whatever ruse the demon puts on. The demon itself would have been summoned by Circle mages to the same part of the Fade the apprentice enters, lured by the promise of a living body waiting for it. The Harrowing is more of a test of will, common sense, and the ability to resist temptation, rather than one of magical ability.
If the apprentice succeeds in resisting the demon, he or she earns the title and trappings of a mage, becoming a full member of the Circle, having proven the necessary strength of will and character to master and use magic.
However, if the apprentice is overwhelmed by the demon - or, some say takes too long to complete the task - he or she is killed by templars, who are standing by to prevent the tragedy of an Abomination. One templar is assigned the killing blow in advance, while the rest keep danger at bay and stand ready to replace the designated slayer if necessary.
Apprentices may volunteer to be made Tranquil instead of undergoing their Harrowing. Since passing the Harrowing theoretically proves that a mage is capable enough to resist possession, The Chantry's law forbids mages who passed their Harrowing from being made Tranquil.
Upon successful completion of his or her Harrowing, a mage is traditionally given a ring of lyrium-infused silver.