In Brave New World, John is well education and eloquent, but moody and violent with his mood swings. As such, he is a Noble Savage, as he is raised on the Savage Reservation. He's got Mommy and daddy issues: he's ashamed of his mother, and on an Christian/existential quest to know who he is and how is like his real father.
In Freudian terms, John is all superego (conscience) or id (desire): there's no middle ground. He ends up dating someone just like his mother (Linda), Lenina. In this way, he suffers from a kind of Oedipal complex. His repression and guilt over his physical relations with Lenina drives him to mania and suicide.
His name comes from the verse drama The Conquest of Granada by John Dryden. Huxley bases him on a combination of Caliban and Alonso from Shakespeare's The Tempest. He is part slave/savage like Caliban and part civilized Utopian like Alonso. After all, the title is taken from a The Tempest, Act V:
O wonder! How many goodly creatures are there here! How beauteous mankind is! O brave new world! That has such people in't!
Christ-like martyr: John is a son who is split between two worlds, the natural and the supernatural. He has two fathers, an earthly one and a mysterious other. In the end, he dies for the sins of others: for us, the readers, and for his father, the Director. He is meant to show us and the Director the extreme effects of both the utopia and the dystopia. In the lighthouse, his body is in the shape similar to that of Christ on the cross.
Byronic Hero: John is an extremist, like Lord Byron the Romantic poet. He is either madly in love or morbidly depressed. There's no moderation or middle course with him. In this way, he is "mad, bad, and dangerous to know." He is wounded by love, exiled for his beliefs, and rebellious against authority. As a tragic hero, he makes mistakes that lead to his death, namely participating in orgy-porgy and taking soma.
To me, John the Savage is best described as a zealot or a true believer -- an idealist. He is someone who believes so completely in the rightness of his values that he cannot bear to live among people who do not share them. And he cannot bear to live after he himself violates his values.
There is no moderation in John. There is no idea of "live and let live." He is a hardliner who sees the world in black and white. This makes him a good character for the book, but it makes me have less affection for him. I would think he could do with a bit of perspective and tolerance. I wonder why he didn't ask to go with Helmholtz to an island...