John is called the savage because he grew up on the reservation in New Mexico where all the "savages" live.  Because of that, his ways are really strange in the eyes of the "civilized" people.

His childhood was totally painful.  All the Indian women hated his mom because she acted (in their opinion) like a whore (she had the values of "civilization").  None of the kids liked him because of that as well.

He feels he is bound by the laws of the society where he grew up, though.  He believes in their morality and the morality he has picked up from Shakespeare plays.

So he is an outcast in two societies, which is why he ends up killing himself, I believe.

Why is the nickname "Savage" for John one of the best examples of irony in the novel Brave New World?

Savage is an adjective expressing exactly the opposite of John's personality: He has a very well-rounded, even sophisticated, mindset, as seen by his knowledge of subjects as diverse as Shakespeare. What's more, he has chivalrous intentions in most circumstances -- he doesn't immediately want to have sexual relations with Lenina, as an example, and believes they should become more personally familiar with each other before such exchanges take place. What's more, he seems very in touch with his sensitive or emotional side, meaning that he lacks savagery. This fact is what makes the "Savage" name ironic.

Why is the nickname "Savage" for John one of the best examples of irony in the novel Brave New World?

In our society, John would actually be considered quite accomplished and cultured. His encyclopedic knowledge of Shakespeare and his romantic notions about life would be considered very attractive. However, in the world of Our Ford, John is considered somewhat crazy because he actually wants to mourn his mother, really get to know Lenina before they jump into bed, and to really feel his emotions instead of smothering them with soma. He represents what would be the best in our society; but in the world of "Our Ford" he is just a curiosity.