Feynriel is an elf-blooded apostate living in Kirkwall.
Background
Feynriel was born to Arianni, a Dalish elf, and a travelling Human Antivan merchant named Vincento. His father abandoned him, forcing Arianni to abandon her clan and raise the boy alone in Kirkwall's Alienage. As a result of his parentage, Feynriel is Elf-blooded.
In his later years, Feynriel began to display magical ability. At first he hid to avoid being sent to the Circle of Magi, but demons began whispering to him in his dreams. Fearing for his life, Arianni sought to bring him to the Circle so that he could receive aid, but Feynriel saw this as a betrayal and fled to seek out the Dalish, who he believed could teach him to control his dreams and his magic. He was captured by slavers while on the run, only to be saved by Hawke.
Involvement
Hawke will receive the quest Wayward Son from Feynriel's mother Arianni in the Lowtown Elven Alienage. He has previously been able to hide from the Templars, but when he starts having nightmares about demons, his mother contacts the Circle out of fear for his life. Feynriel subsequently runs away to remain free, and Hawke has to locate him.
Feynriel returns in a quest in Act 2, Night Terrors, in which Hawke is asked to solve the problem of his nightmares once and for all. Arianni will call Marethari to the Alienage, where she will tell Hawke that Feynriel is a "Somniari," or "Dreamer," an exceptional skill making him able to enter and change the Fade at will, even killing people within it. As such, it would be extremely dangerous if he were to become an abomination. Hawke must enter the Fade and confront Feynriel.
In Act 3, Feynriel is mentioned in the side-quest Who Needs Rescuing? if he is not made Tranquil or possessed.
Wayward Son
The Circle
Hawke can choose to send Feynriel to the Circle of Magi, which will go against his wishes. He will bitterly and reluctantly comply, telling you that you should go and tell Arianni that "she has won." You will be able to see him at the Gallows, where he will remark that it is just as he feared there, and that according to the templars, the only way to stop his dreams is to destroy his mind.
Note that this option will net 
(+5) from Aveline and Varric, in addition to 
Merrill: rivalry (+10) and Anders: rivalry (+15).
The Dalish
Alternatively, Hawke can do as Feynriel asks, and allow him to go to Marethari's clan, where he hopes to control his dreams, and hone his skills as a mage.
This option will net you (+5) from Varric, (+10) from Bethany, (+10) from Anders, and Aveline: rivalry (+5). Choosing to let Feynriel choose what's best for himself (which will be the Dalish) gives you Fenris: rivalry (+10), and telling him you are friends with the Keeper and offering to make sure he has a place with them gives you Fenris: rivalry (+20).
If you go to Sundermount before completing Night Terrors, you may encounter a group of Dalish Hunters and Templars arguing over where Feynriel should go. You can choose to side with the Dalish and fight the Templars Isabela: rivalry (+5), side with the Templars against the Dalish, leave the situation without taking sides Isabela: friendship (+5), or if you investigate both sides of the situation, can choose to "Let Feynriel be." Hawke argues that should Feynriel become an abomination, he will likely kill the Dalish and not the people of Kirkwall, which will satisfy the Templars and resolve the conflict.
Night Terrors
Hawke's actions in the Fade when dealing with Feynriel have a direct effect on his state of mind. The ideal outcome is a stable, non-possessed Feynriel sent to Tevinter; the alternative results in him becoming a "Dreamstalker", which makes him "unstoppable" as Merethari fears.
Torpor the Sloth Demon
In the Fade, Hawke will encounter a Sloth demon, Torpor. It will ask Hawke to be allowed to possess Feynriel in return for a reward. Hawke can either accept the offer, or fight the demon.
If Feynriel has become scared by the demons (which will occur if you do not help him see through the illusions with rational arguments and reason), Torpor will attack Hawke and his party in a rage.
Dreamstalker
Otherwise, he will be possessed by the demon, and in Act 3 you will encounter Grigor, a member of the city guard, in Lowtown at night cornered by Melson and two of his personal guards. Melson has gone crazy and killed several city guards, saying that Feynriel told him that they were thieves. After a fight, Grigor refers to Feynriel as "The Dreamstalker".
Tevinter
If Feynriel is calm and Torpor dead, he will ask Hawke to let him go to the Tevinter Imperium, where he will learn to control his nightmares and master his abilities. He will send you a letter about his time there at the beginning of Act 3. Arianni will be upset that she is unable to say farewell, but understands Feynriel's choice. This result also unlocks the Who Needs Rescuing? quest where Orlanna reveals that they had been communicating in her dreams, and that he used his powers to prevent her from being violated by a group of bandits who had abducted her.
Tranquil
If Feynriel's mind is unstable, Hawke can grant Feynriel's wish and kill him in the Fade, which effectively makes him Tranquil in the real world. Arianni will be devastated by this news and later commits suicide.
Gallery
